Theorem cgsex4g 1369

Description: An implicit substitution inference for 4 general classes.

Hypotheses

Ref	Expression
cgsex4g.1	|- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> ch)
cgsex4g.2	|- (ch -> (ph <-> ps))

Assertion

Ref	Expression
cgsex4g	|- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w(ch /\ ph) <-> ps))

Distinct variable group(s):   x,y,z,w,A   x,B,y,z,w   x,C,y,z,w   x,D,y,z,w   ps,x,y,z,w

Proof of Theorem cgsex4g

Step	Hyp	Ref	Expression
1		cgsex4g.2	. . . . . 6 |- (ch -> (ph <-> ps))
2	1	biimpa 324	. . . . 5 |- ((ch /\ ph) -> ps)
3	2	19.23aivv 953	. . . 4 |- (E.zE.w(ch /\ ph) -> ps)
4	3	19.23aivv 953	. . 3 |- (E.xE.yE.zE.w(ch /\ ph) -> ps)
5	4	a1i 7	. 2 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w(ch /\ ph) -> ps))
6	1	biimprcd 138	. . . . . . 7 |- (ps -> (ch -> ph))
7	6	ancld 246	. . . . . 6 |- (ps -> (ch -> (ch /\ ph)))
8	7	19.22dvv 949	. . . . 5 |- (ps -> (E.zE.wch -> E.zE.w(ch /\ ph)))
9	8	19.22dvv 949	. . . 4 |- (ps -> (E.xE.yE.zE.wch -> E.xE.yE.zE.w(ch /\ ph)))
10		elex 1356	. . . . . . . . 9 |- (A e. R -> E.x x = A)
11		elex 1356	. . . . . . . . 9 |- (B e. S -> E.y y = B)
12	10, 11	anim12i 268	. . . . . . . 8 |- ((A e. R /\ B e. S) -> (E.x x = A /\ E.y y = B))
13		eeanv 980	. . . . . . . 8 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
14	12, 13	sylibr 175	. . . . . . 7 |- ((A e. R /\ B e. S) -> E.xE.y(x = A /\ y = B))
15		elex 1356	. . . . . . . . 9 |- (C e. R -> E.z z = C)
16		elex 1356	. . . . . . . . 9 |- (D e. S -> E.w w = D)
17	15, 16	anim12i 268	. . . . . . . 8 |- ((C e. R /\ D e. S) -> (E.z z = C /\ E.w w = D))
18		eeanv 980	. . . . . . . 8 |- (E.zE.w(z = C /\ w = D) <-> (E.z z = C /\ E.w w = D))
19	17, 18	sylibr 175	. . . . . . 7 |- ((C e. R /\ D e. S) -> E.zE.w(z = C /\ w = D))
20	14, 19	anim12i 268	. . . . . 6 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.y(x = A /\ y = B) /\ E.zE.w(z = C /\ w = D)))
21		ee4anv 982	. . . . . 6 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) <-> (E.xE.y(x = A /\ y = B) /\ E.zE.w(z = C /\ w = D)))
22	20, 21	sylibr 175	. . . . 5 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)))
23		cgsex4g.1	. . . . . . . . 9 |- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> ch)
24	23	19.22i 723	. . . . . . . 8 |- (E.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.wch)
25	24	19.22i 723	. . . . . . 7 |- (E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.zE.wch)
26	25	19.22i 723	. . . . . 6 |- (E.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.yE.zE.wch)
27	26	19.22i 723	. . . . 5 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.xE.yE.zE.wch)
28	22, 27	syl 12	. . . 4 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> E.xE.yE.zE.wch)
29	9, 28	syl5 22	. . 3 |- (ps -> (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> E.xE.yE.zE.w(ch /\ ph)))
30	29	com12 13	. 2 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (ps -> E.xE.yE.zE.w(ch /\ ph)))
31	5, 30	impbid 397	1 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w(ch /\ ph) <-> ps))

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

This theorem is referenced by:  copsex4g 1904  brecop 3242

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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