Theorem spc2gv 2637

Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Hypothesis

Ref	Expression
spc2egv.1	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
spc2gv	|- ((A e. V /\ B e. W) -> (A.xA.yph -> ps))

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

Allowed substitution hints:   ph(x,y)   V(x,y)   W(x,y)

Proof of Theorem spc2gv

Step	Hyp	Ref	Expression
1		elisset 2562	. . . 4 |- (A e. V -> E.x x = A)
2		elisset 2562	. . . 4 |- (B e. W -> E.y y = B)
3	1, 2	anim12i 321	. . 3 |- ((A e. V /\ B e. W) -> (E.x x = A /\ E.y y = B))
4		eeanv 1804	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
5	3, 4	sylibr 137	. 2 |- ((A e. V /\ B e. W) -> E.xE.y(x = A /\ y = B))
6		spc2egv.1	. . . . . 6 |- ((x = A /\ y = B) -> (ph <-> ps))
7	6	biimpcd 148	. . . . 5 |- (ph -> ((x = A /\ y = B) -> ps))
8	7	2alimi 1342	. . . 4 |- (A.xA.yph -> A.xA.y((x = A /\ y = B) -> ps))
9		exim 1487	. . . . 5 |- (A.y((x = A /\ y = B) -> ps) -> (E.y(x = A /\ y = B) -> E.yps))
10	9	alimi 1341	. . . 4 |- (A.xA.y((x = A /\ y = B) -> ps) -> A.x(E.y(x = A /\ y = B) -> E.yps))
11		exim 1487	. . . 4 |- (A.x(E.y(x = A /\ y = B) -> E.yps) -> (E.xE.y(x = A /\ y = B) -> E.xE.yps))
12	8, 10, 11	3syl 17	. . 3 |- (A.xA.yph -> (E.xE.y(x = A /\ y = B) -> E.xE.yps))
13		19.9v 1748	. . . 4 |- (E.xE.yps <-> E.yps)
14		19.9v 1748	. . . 4 |- (E.yps <-> ps)
15	13, 14	bitri 173	. . 3 |- (E.xE.yps <-> ps)
16	12, 15	syl6ib 150	. 2 |- (A.xA.yph -> (E.xE.y(x = A /\ y = B) -> ps))
17	5, 16	syl5com 26	1 |- ((A e. V /\ B e. W) -> (A.xA.yph -> ps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  trel  3852  elovmpt2  5643