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Theorem reuss2 2265
Description: Transfer uniqueness to a smaller subclass.
Assertion
Ref Expression
reuss2 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)
Distinct variable groups:   x,A   x,B
Proof of Theorem reuss2
StepHypRef Expression
1 prth 554 . . . . . . . . . . . . . 14 |- (((x e. A -> x e. B) /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
2 ssel 2053 . . . . . . . . . . . . . 14 |- (A (_ B -> (x e. A -> x e. B))
31, 2sylan 448 . . . . . . . . . . . . 13 |- ((A (_ B /\ (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps)))
43exp4b 379 . . . . . . . . . . . 12 |- (A (_ B -> ((ph -> ps) -> (x e. A -> (ph -> (x e. B /\ ps)))))
54com23 32 . . . . . . . . . . 11 |- (A (_ B -> (x e. A -> ((ph -> ps) -> (ph -> (x e. B /\ ps)))))
65a2d 13 . . . . . . . . . 10 |- (A (_ B -> ((x e. A -> (ph -> ps)) -> (x e. A -> (ph -> (x e. B /\ ps)))))
76imp4a 364 . . . . . . . . 9 |- (A (_ B -> ((x e. A -> (ph -> ps)) -> ((x e. A /\ ph) -> (x e. B /\ ps))))
8719.20dv 1284 . . . . . . . 8 |- (A (_ B -> (A.x(x e. A -> (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps))))
98imp 350 . . . . . . 7 |- ((A (_ B /\ A.x(x e. A -> (ph -> ps))) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
10 df-ral 1641 . . . . . . 7 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
119, 10sylan2b 452 . . . . . 6 |- ((A (_ B /\ A.x e. A (ph -> ps)) -> A.x((x e. A /\ ph) -> (x e. B /\ ps)))
12 euimmo 1413 . . . . . 6 |- (A.x((x e. A /\ ph) -> (x e. B /\ ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
1311, 12syl 10 . . . . 5 |- ((A (_ B /\ A.x e. A (ph -> ps)) -> (E!x(x e. B /\ ps) -> E*x(x e. A /\ ph)))
14 eu5 1402 . . . . . . 7 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)))
1514biimpr 152 . . . . . 6 |- ((E.x(x e. A /\ ph) /\ E*x(x e. A /\ ph)) -> E!x(x e. A /\ ph))
1615ex 373 . . . . 5 |- (E.x(x e. A /\ ph) -> (E*x(x e. A /\ ph) -> E!x(x e. A /\ ph)))
1713, 16syl9 57 . . . 4 |- ((A (_ B /\ A.x e. A (ph -> ps)) -> (E.x(x e. A /\ ph) -> (E!x(x e. B /\ ps) -> E!x(x e. A /\ ph))))
1817imp32 363 . . 3 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps))) -> E!x(x e. A /\ ph))
19 df-reu 1643 . . 3 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2018, 19sylibr 200 . 2 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps))) -> E!x e. A ph)
21 df-rex 1642 . . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
22 df-reu 1643 . . 3 |- (E!x e. B ps <-> E!x(x e. B /\ ps))
2321, 22anbi12i 481 . 2 |- ((E.x e. A ph /\ E!x e. B ps) <-> (E.x(x e. A /\ ph) /\ E!x(x e. B /\ ps)))
2420, 23sylan2b 452 1 |- (((A (_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   e. wcel 955  E.wex 977  E!weu 1373  E*wmo 1374  A.wral 1637  E.wrex 1638  E!wreu 1639   (_ wss 2037
This theorem is referenced by:  reuss 2266  reuun1 2267  reuuniss2 2881  grpidinv2 7994  grpinv 8003
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-reu 1643  df-in 2041  df-ss 2043
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