Theorem cgsex2g 2584

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Hypotheses

Ref	Expression
cgsex2g.1	|- ((x = A /\ y = B) -> ch)
cgsex2g.2	|- (ch -> (ph <-> ps))

Assertion

Ref	Expression
cgsex2g	|- ((A e. V /\ B e. W) -> (E.xE.y(ch /\ ph) <-> ps))

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

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

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . 4 |- (ch -> (ph <-> ps))
2	1	biimpa 280	. . 3 |- ((ch /\ ph) -> ps)
3	2	exlimivv 1773	. 2 |- (E.xE.y(ch /\ ph) -> ps)
4		elisset 2562	. . . . . 6 |- (A e. V -> E.x x = A)
5		elisset 2562	. . . . . 6 |- (B e. W -> E.y y = B)
6	4, 5	anim12i 321	. . . . 5 |- ((A e. V /\ B e. W) -> (E.x x = A /\ E.y y = B))
7		eeanv 1804	. . . . 5 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
8	6, 7	sylibr 137	. . . 4 |- ((A e. V /\ B e. W) -> E.xE.y(x = A /\ y = B))
9		cgsex2g.1	. . . . 5 |- ((x = A /\ y = B) -> ch)
10	9	2eximi 1489	. . . 4 |- (E.xE.y(x = A /\ y = B) -> E.xE.ych)
11	8, 10	syl 14	. . 3 |- ((A e. V /\ B e. W) -> E.xE.ych)
12	1	biimprcd 149	. . . . 5 |- (ps -> (ch -> ph))
13	12	ancld 308	. . . 4 |- (ps -> (ch -> (ch /\ ph)))
14	13	2eximdv 1759	. . 3 |- (ps -> (E.xE.ych -> E.xE.y(ch /\ ph)))
15	11, 14	syl5com 26	. 2 |- ((A e. V /\ B e. W) -> (ps -> E.xE.y(ch /\ ph)))
16	3, 15	impbid2 131	1 |- ((A e. V /\ B e. W) -> (E.xE.y(ch /\ ph) <-> ps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = 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: (None)