Theorem elabreximd 29348

Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)

Hypotheses

Ref	Expression
elabreximd.1	|- F/ x ph
elabreximd.2	|- F/ x ch
elabreximd.3	|- ( A = B -> ( ch <-> ps ) )
elabreximd.4	|- ( ph -> A e. V )
elabreximd.5	|- ( ( ph /\ x e. C ) -> ps )

Assertion

Ref	Expression
elabreximd	|- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch )

Distinct variable groups:   x, y, A   y, B   y, C

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, y)   B( x)   C( x)   V( x, y)

Proof of Theorem elabreximd

Step	Hyp	Ref	Expression
1		elabreximd.4	. . . 4 |- ( ph -> A e. V )
2		eqeq1 2626	. . . . . 6 |- ( y = A -> ( y = B <-> A = B ) )
3	2	rexbidv 3052	. . . . 5 |- ( y = A -> ( E. x e. C y = B <-> E. x e. C A = B ) )
4	3	elabg 3351	. . . 4 |- ( A e. V -> ( A e. { y | E. x e. C y = B } <-> E. x e. C A = B ) )
5	1, 4	syl 17	. . 3 |- ( ph -> ( A e. { y | E. x e. C y = B } <-> E. x e. C A = B ) )
6	5	biimpa 501	. 2 |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> E. x e. C A = B )
7		elabreximd.1	. . . 4 |- F/ x ph
8		elabreximd.2	. . . 4 |- F/ x ch
9		simpr 477	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ A = B ) -> A = B )
10		elabreximd.5	. . . . . . 7 |- ( ( ph /\ x e. C ) -> ps )
11	10	adantr 481	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ A = B ) -> ps )
12		elabreximd.3	. . . . . . 7 |- ( A = B -> ( ch <-> ps ) )
13	12	biimpar 502	. . . . . 6 |- ( ( A = B /\ ps ) -> ch )
14	9, 11, 13	syl2anc 693	. . . . 5 |- ( ( ( ph /\ x e. C ) /\ A = B ) -> ch )
15	14	exp31 630	. . . 4 |- ( ph -> ( x e. C -> ( A = B -> ch ) ) )
16	7, 8, 15	rexlimd 3026	. . 3 |- ( ph -> ( E. x e. C A = B -> ch ) )
17	16	imp 445	. 2 |- ( ( ph /\ E. x e. C A = B ) -> ch )
18	6, 17	syldan 487	1 |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   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-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  elabreximdv  29349  abrexss  29350  disjabrex  29395  disjabrexf  29396