Theorem cgsex2g 1368

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 group(s):   x,y,ps   x,A,y   x,B,y

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . . 5 |- (ch -> (ph <-> ps))
2	1	biimpa 324	. . . 4 |- ((ch /\ ph) -> ps)
3	2	19.23aivv 953	. . 3 |- (E.xE.y(ch /\ ph) -> ps)
4	3	a1i 7	. 2 |- ((A e. C /\ B e. D) -> (E.xE.y(ch /\ ph) -> ps))
5	1	biimprcd 138	. . . . . 6 |- (ps -> (ch -> ph))
6	5	ancld 246	. . . . 5 |- (ps -> (ch -> (ch /\ ph)))
7	6	19.22dvv 949	. . . 4 |- (ps -> (E.xE.ych -> E.xE.y(ch /\ ph)))
8		elex 1356	. . . . . . 7 |- (A e. C -> E.x x = A)
9		elex 1356	. . . . . . 7 |- (B e. D -> E.y y = B)
10	8, 9	anim12i 268	. . . . . 6 |- ((A e. C /\ B e. D) -> (E.x x = A /\ E.y y = B))
11		eeanv 980	. . . . . 6 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
12	10, 11	sylibr 175	. . . . 5 |- ((A e. C /\ B e. D) -> E.xE.y(x = A /\ y = B))
13		cgsex2g.1	. . . . . . 7 |- ((x = A /\ y = B) -> ch)
14	13	19.22i 723	. . . . . 6 |- (E.y(x = A /\ y = B) -> E.ych)
15	14	19.22i 723	. . . . 5 |- (E.xE.y(x = A /\ y = B) -> E.xE.ych)
16	12, 15	syl 12	. . . 4 |- ((A e. C /\ B e. D) -> E.xE.ych)
17	7, 16	syl5 22	. . 3 |- (ps -> ((A e. C /\ B e. D) -> E.xE.y(ch /\ ph)))
18	17	com12 13	. 2 |- ((A e. C /\ B e. D) -> (ps -> E.xE.y(ch /\ ph)))
19	4, 18	impbid 397	1 |- ((A e. C /\ B e. D) -> (E.xE.y(ch /\ ph) <-> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092

This theorem is referenced by:  distrlem5pr 3925

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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