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Theorem 2reuswap 1927
Description: A condition allowing swap of uniqueness and existential quantifiers.
Assertion
Ref Expression
2reuswap |- (A.x e. A E*y(y e. A /\ ph) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))
Distinct variable group:   x,y,A
Proof of Theorem 2reuswap
StepHypRef Expression
1 df-ral 1641 . . 3 |- (A.x e. A E*y(y e. A /\ ph) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
2 moanimv 1422 . . . 4 |- (E*y(x e. A /\ (y e. A /\ ph)) <-> (x e. A -> E*y(y e. A /\ ph)))
32albii 996 . . 3 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) <-> A.x(x e. A -> E*y(y e. A /\ ph)))
41, 3bitr4 176 . 2 |- (A.x e. A E*y(y e. A /\ ph) <-> A.xE*y(x e. A /\ (y e. A /\ ph)))
5 2euswap 1438 . . 3 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) -> (E!xE.y(x e. A /\ (y e. A /\ ph)) -> E!yE.x(x e. A /\ (y e. A /\ ph))))
6 df-reu 1643 . . . 4 |- (E!x e. A E.y e. A ph <-> E!x(x e. A /\ E.y e. A ph))
7 df-rex 1642 . . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> E.y(y e. A /\ (x e. A /\ ph)))
8 r19.42v 1756 . . . . . 6 |- (E.y e. A (x e. A /\ ph) <-> (x e. A /\ E.y e. A ph))
9 an12 483 . . . . . . 7 |- ((y e. A /\ (x e. A /\ ph)) <-> (x e. A /\ (y e. A /\ ph)))
109exbii 1047 . . . . . 6 |- (E.y(y e. A /\ (x e. A /\ ph)) <-> E.y(x e. A /\ (y e. A /\ ph)))
117, 8, 103bitr3 181 . . . . 5 |- ((x e. A /\ E.y e. A ph) <-> E.y(x e. A /\ (y e. A /\ ph)))
1211eubii 1380 . . . 4 |- (E!x(x e. A /\ E.y e. A ph) <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
136, 12bitr 173 . . 3 |- (E!x e. A E.y e. A ph <-> E!xE.y(x e. A /\ (y e. A /\ ph)))
14 df-reu 1643 . . . 4 |- (E!y e. A E.x e. A ph <-> E!y(y e. A /\ E.x e. A ph))
15 r19.42v 1756 . . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> (y e. A /\ E.x e. A ph))
16 df-rex 1642 . . . . . 6 |- (E.x e. A (y e. A /\ ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
1715, 16bitr3 175 . . . . 5 |- ((y e. A /\ E.x e. A ph) <-> E.x(x e. A /\ (y e. A /\ ph)))
1817eubii 1380 . . . 4 |- (E!y(y e. A /\ E.x e. A ph) <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
1914, 18bitr 173 . . 3 |- (E!y e. A E.x e. A ph <-> E!yE.x(x e. A /\ (y e. A /\ ph)))
205, 13, 193imtr4g 551 . 2 |- (A.xE*y(x e. A /\ (y e. A /\ ph)) -> (E!x e. A E.y e. A ph -> E!y e. A E.x e. A ph))
214, 20sylbi 199 1 |- (A.x e. A E*y(y e. A /\ ph) -> (E!x e. A E.y e. A ph -> E!y e. A 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
This theorem is referenced by:  reuxfr2 2893
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
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-ral 1641  df-rex 1642  df-reu 1643
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