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Theorem zfrep4 2691
Description: A version of Replacement using class abstractions.
Hypotheses
Ref Expression
zfrep4.1 |- {x | ph} e. V
zfrep4.2 |- (ph -> E.zA.y(ps -> y = z))
Assertion
Ref Expression
zfrep4 |- {y | E.x(ph /\ ps)} e. V
Distinct variable groups:   ph,y,z   ps,z   x,y,z
Proof of Theorem zfrep4
StepHypRef Expression
1 abid 1458 . . . . 5 |- (x e. {x | ph} <-> ph)
21anbi1i 480 . . . 4 |- ((x e. {x | ph} /\ ps) <-> (ph /\ ps))
32exbii 1047 . . 3 |- (E.x(x e. {x | ph} /\ ps) <-> E.x(ph /\ ps))
43abbii 1567 . 2 |- {y | E.x(x e. {x | ph} /\ ps)} = {y | E.x(ph /\ ps)}
5 hbab1 1459 . . . . 5 |- (y e. {x | ph} -> A.x y e. {x | ph})
6 zfrep4.1 . . . . 5 |- {x | ph} e. V
7 zfrep4.2 . . . . . 6 |- (ph -> E.zA.y(ps -> y = z))
81, 7sylbi 199 . . . . 5 |- (x e. {x | ph} -> E.zA.y(ps -> y = z))
95, 6, 8zfrepclf 2689 . . . 4 |- E.zA.y(y e. z <-> E.x(x e. {x | ph} /\ ps))
10 abeq2 1560 . . . . 5 |- (z = {y | E.x(x e. {x | ph} /\ ps)} <-> A.y(y e. z <-> E.x(x e. {x | ph} /\ ps)))
1110exbii 1047 . . . 4 |- (E.z z = {y | E.x(x e. {x | ph} /\ ps)} <-> E.zA.y(y e. z <-> E.x(x e. {x | ph} /\ ps)))
129, 11mpbir 190 . . 3 |- E.z z = {y | E.x(x e. {x | ph} /\ ps)}
1312issetri 1807 . 2 |- {y | E.x(x e. {x | ph} /\ ps)} e. V
144, 13eqeltrr 1537 1 |- {y | E.x(ph /\ ps)} e. V
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  Vcvv 1802
This theorem is referenced by:  zfpair 2767
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-12 965  ax-14 967  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  ax-rep 2683
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803
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