Theorem ceqsrex2v 2975

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)

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

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

Proof of Theorem ceqsrex2v

Step	Hyp	Ref	Expression
1		anass 630	. . . . . 6 |- (((x = A /\ y = B) /\ ph) <-> (x = A /\ (y = B /\ ph)))
2	1	rexbii 2640	. . . . 5 |- (E.y e. D ((x = A /\ y = B) /\ ph) <-> E.y e. D (x = A /\ (y = B /\ ph)))
3		r19.42v 2766	. . . . 5 |- (E.y e. D (x = A /\ (y = B /\ ph)) <-> (x = A /\ E.y e. D (y = B /\ ph)))
4	2, 3	bitri 240	. . . 4 |- (E.y e. D ((x = A /\ y = B) /\ ph) <-> (x = A /\ E.y e. D (y = B /\ ph)))
5	4	rexbii 2640	. . 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)))
6		ceqsrex2v.1	. . . . . 6 |- (x = A -> (ph <-> ps))
7	6	anbi2d 684	. . . . 5 |- (x = A -> ((y = B /\ ph) <-> (y = B /\ ps)))
8	7	rexbidv 2636	. . . 4 |- (x = A -> (E.y e. D (y = B /\ ph) <-> E.y e. D (y = B /\ ps)))
9	8	ceqsrexv 2973	. . 3 |- (A e. C -> (E.x e. C (x = A /\ E.y e. D (y = B /\ ph)) <-> E.y e. D (y = B /\ ps)))
10	5, 9	syl5bb 248	. 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 2973	. 2 |- (B e. D -> (E.y e. D (y = B /\ ps) <-> ch))
13	10, 12	sylan9bb 680	1 |- ((A e. C /\ B e. D) -> (E.x e. C E.y e. D ((x = A /\ y = B) /\ ph) <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862

This theorem is referenced by: (None)