Theorem cgsex4g 1833

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 groups:   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	. . . . 5 |- (ch -> (ph <-> ps))
2	1	biimpa 416	. . . 4 |- ((ch /\ ph) -> ps)
3	2	19.23aivv 1296	. . 3 |- (E.zE.w(ch /\ ph) -> ps)
4	3	19.23aivv 1296	. 2 |- (E.xE.yE.zE.w(ch /\ ph) -> ps)
5	1	biimprcd 156	. . . . . 6 |- (ps -> (ch -> ph))
6	5	ancld 298	. . . . 5 |- (ps -> (ch -> (ch /\ ph)))
7	6	19.22dvv 1292	. . . 4 |- (ps -> (E.zE.wch -> E.zE.w(ch /\ ph)))
8	7	19.22dvv 1292	. . 3 |- (ps -> (E.xE.yE.zE.wch -> E.xE.yE.zE.w(ch /\ ph)))
9		elex 1819	. . . . . . . 8 |- (A e. R -> E.x x = A)
10		elex 1819	. . . . . . . 8 |- (B e. S -> E.y y = B)
11	9, 10	anim12i 333	. . . . . . 7 |- ((A e. R /\ B e. S) -> (E.x x = A /\ E.y y = B))
12		eeanv 1323	. . . . . . 7 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
13	11, 12	sylibr 200	. . . . . 6 |- ((A e. R /\ B e. S) -> E.xE.y(x = A /\ y = B))
14		elex 1819	. . . . . . . 8 |- (C e. R -> E.z z = C)
15		elex 1819	. . . . . . . 8 |- (D e. S -> E.w w = D)
16	14, 15	anim12i 333	. . . . . . 7 |- ((C e. R /\ D e. S) -> (E.z z = C /\ E.w w = D))
17		eeanv 1323	. . . . . . 7 |- (E.zE.w(z = C /\ w = D) <-> (E.z z = C /\ E.w w = D))
18	16, 17	sylibr 200	. . . . . 6 |- ((C e. R /\ D e. S) -> E.zE.w(z = C /\ w = D))
19	13, 18	anim12i 333	. . . . 5 |- (((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)))
20		ee4anv 1325	. . . . 5 |- (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)))
21	19, 20	sylibr 200	. . . 4 |- (((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)))
22		cgsex4g.1	. . . . . 6 |- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> ch)
23	22	19.22i2 1041	. . . . 5 |- (E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.zE.wch)
24	23	19.22i2 1041	. . . 4 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.xE.yE.zE.wch)
25	21, 24	syl 10	. . 3 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> E.xE.yE.zE.wch)
26	8, 25	syl5com 52	. 2 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (ps -> E.xE.yE.zE.w(ch /\ ph)))
27	4, 26	impbid2 518	1 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w(ch /\ ph) <-> ps))

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

This theorem is referenced by:  copsex4g 2794  brecop 4306

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

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