Theorem zfrep4 2775

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

Step	Hyp	Ref	Expression
1		abid 1507	. . . . 5 |- (x e. {x | ph} <-> ph)
2	1	anbi1i 484	. . . 4 |- ((x e. {x | ph} /\ ps) <-> (ph /\ ps))
3	2	exbii 1087	. . 3 |- (E.x(x e. {x | ph} /\ ps) <-> E.x(ph /\ ps))
4	3	abbii 1618	. 2 |- {y | E.x(x e. {x | ph} /\ ps)} = {y | E.x(ph /\ ps)}
5		hbab1 1508	. . . . 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))
8	1, 7	sylbi 197	. . . . 5 |- (x e. {x | ph} -> E.zA.y(ps -> y = z))
9	5, 6, 8	zfrepclf 2773	. . . 4 |- E.zA.y(y e. z <-> E.x(x e. {x | ph} /\ ps))
10		abeq2 1611	. . . . 5 |- (z = {y | E.x(x e. {x | ph} /\ ps)} <-> A.y(y e. z <-> E.x(x e. {x | ph} /\ ps)))
11	10	exbii 1087	. . . 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)))
12	9, 11	mpbir 188	. . 3 |- E.z z = {y | E.x(x e. {x | ph} /\ ps)}
13	12	issetri 1862	. 2 |- {y | E.x(x e. {x | ph} /\ ps)} e. V
14	4, 13	eqeltrri 1588	1 |- {y | E.x(ph /\ ps)} e. V

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  E.wex 1016  {cab 1505  Vcvv 1857

This theorem is referenced by:  zfpair 2853

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858

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