Theorem elabreximd 6329

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 2241	. . . . . 6 |- ( y = A -> ( y = B <-> A = B ) )
3	2	rexbidv 2545	. . . . 5 |- ( y = A -> ( E. x e. C y = B <-> E. x e. C A = B ) )
4	3	elabg 2966	. . . 4 |- ( A e. V -> ( A e. { y | E. x e. C y = B } <-> E. x e. C A = B ) )
5	1, 4	syl 14	. . 3 |- ( ph -> ( A e. { y | E. x e. C y = B } <-> E. x e. C A = B ) )
6	5	biimpa 296	. 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 110	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ A = B ) -> A = B )
10		elabreximd.5	. . . . . . 7 |- ( ( ph /\ x e. C ) -> ps )
11	10	adantr 276	. . . . . 6 |- ( ( ( ph /\ x e. C ) /\ A = B ) -> ps )
12		elabreximd.3	. . . . . . 7 |- ( A = B -> ( ch <-> ps ) )
13	12	biimpar 297	. . . . . 6 |- ( ( A = B /\ ps ) -> ch )
14	9, 11, 13	syl2anc 411	. . . . 5 |- ( ( ( ph /\ x e. C ) /\ A = B ) -> ch )
15	14	exp31 364	. . . 4 |- ( ph -> ( x e. C -> ( A = B -> ch ) ) )
16	7, 8, 15	rexlimd 2659	. . 3 |- ( ph -> ( E. x e. C A = B -> ch ) )
17	16	imp 124	. 2 |- ( ( ph /\ E. x e. C A = B ) -> ch )
18	6, 17	syldan 282	1 |- ( ( ph /\ A e. { y | E. x e. C y = B } ) -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   F/wnf 1509   e. wcel 2205   {cab 2220   E.wrex 2523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817

This theorem is referenced by:  elabreximdv  6330  abrexss  6331