Theorem zfrep4 2669

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 1442	. . . . 5 |- (x e. {x | ph} <-> ph)
2	1	anbi1i 480	. . . 4 |- ((x e. {x | ph} /\ ps) <-> (ph /\ ps))
3	2	exbii 1027	. . 3 |- (E.x(x e. {x | ph} /\ ps) <-> E.x(ph /\ ps))
4	3	abbii 1551	. 2 |- {y | E.x(x e. {x | ph} /\ ps)} = {y | E.x(ph /\ ps)}
5		hbab1 1443	. . . . 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 199	. . . . 5 |- (x e. {x | ph} -> E.zA.y(ps -> y = z))
9	5, 6, 8	zfrepclf 2667	. . . 4 |- E.zA.y(y e. z <-> E.x(x e. {x | ph} /\ ps))
10		abeq2 1544	. . . . 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 1027	. . . 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 190	. . 3 |- E.z z = {y | E.x(x e. {x | ph} /\ ps)}
13	12	issetri 1791	. 2 |- {y | E.x(x e. {x | ph} /\ ps)} e. V
14	4, 13	eqeltrr 1521	1 |- {y | E.x(ph /\ ps)} e. V

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  {cab 1440  Vcvv 1786

This theorem is referenced by:  zfpair 2745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-14 1108  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787

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