Theorem rexab2 3373

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- ( x = y -> ( ps <-> ch ) )

Assertion

Ref	Expression
rexab2	|- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) )

Distinct variable groups:   x, y   ch, x   ph, x   ps, y

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

Proof of Theorem rexab2

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 |- ( E. x e. { y | ph } ps <-> E. x ( x e. { y | ph } /\ ps ) )
2		nfsab1 2612	. . . 4 |- F/ y x e. { y | ph }
3		nfv 1843	. . . 4 |- F/ y ps
4	2, 3	nfan 1828	. . 3 |- F/ y ( x e. { y | ph } /\ ps )
5		nfv 1843	. . 3 |- F/ x ( ph /\ ch )
6		eleq1 2689	. . . . 5 |- ( x = y -> ( x e. { y | ph } <-> y e. { y | ph } ) )
7		abid 2610	. . . . 5 |- ( y e. { y | ph } <-> ph )
8	6, 7	syl6bb 276	. . . 4 |- ( x = y -> ( x e. { y | ph } <-> ph ) )
9		ralab2.1	. . . 4 |- ( x = y -> ( ps <-> ch ) )
10	8, 9	anbi12d 747	. . 3 |- ( x = y -> ( ( x e. { y | ph } /\ ps ) <-> ( ph /\ ch ) ) )
11	4, 5, 10	cbvex 2272	. 2 |- ( E. x ( x e. { y | ph } /\ ps ) <-> E. y ( ph /\ ch ) )
12	1, 11	bitri 264	1 |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913

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

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-rex 2918

This theorem is referenced by:  rexrab2  3374  tmdgsum2  21900  clrellem  37929  brtrclfv2  38019