Theorem cgsex2g 2892

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 470	. . 3 |- ((ch /\ ph) -> ps)
3	2	exlimivv 1635	. 2 |- (E.xE.y(ch /\ ph) -> ps)
4		elisset 2870	. . . . . 6 |- (A e. V -> E.x x = A)
5		elisset 2870	. . . . . 6 |- (B e. W -> E.y y = B)
6	4, 5	anim12i 549	. . . . 5 |- ((A e. V /\ B e. W) -> (E.x x = A /\ E.y y = B))
7		eeanv 1913	. . . . 5 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
8	6, 7	sylibr 203	. . . 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 1577	. . . 4 |- (E.xE.y(x = A /\ y = B) -> E.xE.ych)
11	8, 10	syl 15	. . 3 |- ((A e. V /\ B e. W) -> E.xE.ych)
12	1	biimprcd 216	. . . . 5 |- (ps -> (ch -> ph))
13	12	ancld 536	. . . 4 |- (ps -> (ch -> (ch /\ ph)))
14	13	2eximdv 1624	. . 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 195	1 |- ((A e. V /\ B e. W) -> (E.xE.y(ch /\ ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710

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-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)