Theorem zfrep4 4779

Description: A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)

Hypotheses

Ref	Expression
zfrep4.1	|- { x | ph } e. _V
zfrep4.2	|- ( ph -> E. z A. 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

Allowed substitution hints:   ph( x)   ps( x, y)

Proof of Theorem zfrep4

Step	Hyp	Ref	Expression
1		abid 2610	. . . . 5 |- ( x e. { x | ph } <-> ph )
2	1	anbi1i 731	. . . 4 |- ( ( x e. { x | ph } /\ ps ) <-> ( ph /\ ps ) )
3	2	exbii 1774	. . 3 |- ( E. x ( x e. { x | ph } /\ ps ) <-> E. x ( ph /\ ps ) )
4	3	abbii 2739	. 2 |- { y | E. x ( x e. { x | ph } /\ ps ) } = { y | E. x ( ph /\ ps ) }
5		nfab1 2766	. . . . 5 |- F/_ x { x | ph }
6		zfrep4.1	. . . . 5 |- { x | ph } e. _V
7		zfrep4.2	. . . . . 6 |- ( ph -> E. z A. y ( ps -> y = z ) )
8	1, 7	sylbi 207	. . . . 5 |- ( x e. { x | ph } -> E. z A. y ( ps -> y = z ) )
9	5, 6, 8	zfrepclf 4777	. . . 4 |- E. z A. y ( y e. z <-> E. x ( x e. { x | ph } /\ ps ) )
10		abeq2 2732	. . . . 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 1774	. . . 4 |- ( E. z z = { y | E. x ( x e. { x | ph } /\ ps ) } <-> E. z A. y ( y e. z <-> E. x ( x e. { x | ph } /\ ps ) ) )
12	9, 11	mpbir 221	. . 3 |- E. z z = { y | E. x ( x e. { x | ph } /\ ps ) }
13	12	issetri 3210	. 2 |- { y | E. x ( x e. { x | ph } /\ ps ) } e. _V
14	4, 13	eqeltrri 2698	1 |- { y | E. x ( ph /\ ps ) } e. _V

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  zfpair  4904  cshwsexa  13570