Theorem cgsex2g 1832

Description: Implicit substitution inference for general classes.

Hypotheses

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

Assertion

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

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

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . 4 |- (ch -> (ph <-> ps))
2	1	biimpa 416	. . 3 |- ((ch /\ ph) -> ps)
3	2	19.23aivv 1296	. 2 |- (E.xE.y(ch /\ ph) -> ps)
4	1	biimprcd 156	. . . . 5 |- (ps -> (ch -> ph))
5	4	ancld 298	. . . 4 |- (ps -> (ch -> (ch /\ ph)))
6	5	19.22dvv 1292	. . 3 |- (ps -> (E.xE.ych -> E.xE.y(ch /\ ph)))
7		elex 1819	. . . . . 6 |- (A e. C -> E.x x = A)
8		elex 1819	. . . . . 6 |- (B e. D -> E.y y = B)
9	7, 8	anim12i 333	. . . . 5 |- ((A e. C /\ B e. D) -> (E.x x = A /\ E.y y = B))
10		eeanv 1323	. . . . 5 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
11	9, 10	sylibr 200	. . . 4 |- ((A e. C /\ B e. D) -> E.xE.y(x = A /\ y = B))
12		cgsex2g.1	. . . . 5 |- ((x = A /\ y = B) -> ch)
13	12	19.22i2 1041	. . . 4 |- (E.xE.y(x = A /\ y = B) -> E.xE.ych)
14	11, 13	syl 10	. . 3 |- ((A e. C /\ B e. D) -> E.xE.ych)
15	6, 14	syl5com 52	. 2 |- ((A e. C /\ B e. D) -> (ps -> E.xE.y(ch /\ ph)))
16	3, 15	impbid2 518	1 |- ((A e. C /\ B e. D) -> (E.xE.y(ch /\ ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  distrlem5pr 5131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

Copyright terms: Public domain