Theorem ceqsrex2v 1893

Description: Elimination of a restricted existential quantifier, using implicit substitution.

Hypotheses

Ref	Expression
ceqsrex2v.1	|- (x = A -> (ph <-> ps))
ceqsrex2v.2	|- (y = B -> (ps <-> ch))

Assertion

Ref	Expression
ceqsrex2v	|- ((A e. C /\ B e. D) -> (E.x e. C E.y e. D ((x = A /\ y = B) /\ ph) <-> ch))

Distinct variable groups:   x,y,A   x,B,y   x,C   x,D,y   ps,x   ch,y

Proof of Theorem ceqsrex2v

Step	Hyp	Ref	Expression
1		ceqsrex2v.1	. . . . . 6 |- (x = A -> (ph <-> ps))
2	1	anbi2d 618	. . . . 5 |- (x = A -> ((y = B /\ ph) <-> (y = B /\ ps)))
3	2	rexbidv 1667	. . . 4 |- (x = A -> (E.y e. D (y = B /\ ph) <-> E.y e. D (y = B /\ ps)))
4	3	ceqsrexv 1892	. . 3 |- (A e. C -> (E.x e. C (x = A /\ E.y e. D (y = B /\ ph)) <-> E.y e. D (y = B /\ ps)))
5		anass 441	. . . . . 6 |- (((x = A /\ y = B) /\ ph) <-> (x = A /\ (y = B /\ ph)))
6	5	rexbii 1671	. . . . 5 |- (E.y e. D ((x = A /\ y = B) /\ ph) <-> E.y e. D (x = A /\ (y = B /\ ph)))
7		r19.42v 1767	. . . . 5 |- (E.y e. D (x = A /\ (y = B /\ ph)) <-> (x = A /\ E.y e. D (y = B /\ ph)))
8	6, 7	bitr 173	. . . 4 |- (E.y e. D ((x = A /\ y = B) /\ ph) <-> (x = A /\ E.y e. D (y = B /\ ph)))
9	8	rexbii 1671	. . 3 |- (E.x e. C E.y e. D ((x = A /\ y = B) /\ ph) <-> E.x e. C (x = A /\ E.y e. D (y = B /\ ph)))
10	4, 9	syl5bb 534	. 2 |- (A e. C -> (E.x e. C E.y e. D ((x = A /\ y = B) /\ ph) <-> E.y e. D (y = B /\ ps)))
11		ceqsrex2v.2	. . 3 |- (y = B -> (ps <-> ch))
12	11	ceqsrexv 1892	. 2 |- (B e. D -> (E.y e. D (y = B /\ ps) <-> ch))
13	10, 12	sylan9bb 542	1 |- ((A e. C /\ B e. D) -> (E.x e. C E.y e. D ((x = A /\ y = B) /\ ph) <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wrex 1649

This theorem is referenced by:  brdom7disj 4814  brdom6disj 4815  dffsum 6998

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-v 1815

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