Theorem zfrep4 1479

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 group(s):   ph,y,z   ps,z   x,y,z

Proof of Theorem zfrep4

Step	Hyp	Ref	Expression
1		abid 1094	. . . . 5 |- (x e. {x | ph} <-> ph)
2	1	anbi1i 368	. . . 4 |- ((x e. {x | ph} /\ ps) <-> (ph /\ ps))
3	2	biex 733	. . 3 |- (E.x(x e. {x | ph} /\ ps) <-> E.x(ph /\ ps))
4	3	biabi 1181	. 2 |- {y | E.x(x e. {x | ph} /\ ps)} = {y | E.x(ph /\ ps)}
5		hbab1 1095	. . . . 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 174	. . . . 5 |- (x e. {x | ph} -> E.zA.y(ps -> y = z))
9	5, 6, 8	zfrepclf 1477	. . . 4 |- E.zA.y(y e. z <-> E.x(x e. {x | ph} /\ ps))
10		cleqab 1174	. . . . 5 |- (z = {y | E.x(x e. {x | ph} /\ ps)} <-> A.y(y e. z <-> E.x(x e. {x | ph} /\ ps)))
11	10	biex 733	. . . 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 165	. . 3 |- E.z z = {y | E.x(x e. {x | ph} /\ ps)}
13	12	issetri 1353	. 2 |- {y | E.x(x e. {x | ph} /\ ps)} e. V
14	4, 13	eqeltrr 1160	1 |- {y | E.x(ph /\ ps)} e. V

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  zfpair 1891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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