Theorem cgsex4g 2893

Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)

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

Allowed substitution hints:   ph(x,y,z,w)   ch(x,y,z,w)   R(x,y,z,w)   S(x,y,z,w)

Proof of Theorem cgsex4g

Step	Hyp	Ref	Expression
1		cgsex4g.2	. . . . 5 |- (ch -> (ph <-> ps))
2	1	biimpa 470	. . . 4 |- ((ch /\ ph) -> ps)
3	2	exlimivv 1635	. . 3 |- (E.zE.w(ch /\ ph) -> ps)
4	3	exlimivv 1635	. 2 |- (E.xE.yE.zE.w(ch /\ ph) -> ps)
5		elisset 2870	. . . . . . . 8 |- (A e. R -> E.x x = A)
6		elisset 2870	. . . . . . . 8 |- (B e. S -> E.y y = B)
7	5, 6	anim12i 549	. . . . . . 7 |- ((A e. R /\ B e. S) -> (E.x x = A /\ E.y y = B))
8		eeanv 1913	. . . . . . 7 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
9	7, 8	sylibr 203	. . . . . 6 |- ((A e. R /\ B e. S) -> E.xE.y(x = A /\ y = B))
10		elisset 2870	. . . . . . . 8 |- (C e. R -> E.z z = C)
11		elisset 2870	. . . . . . . 8 |- (D e. S -> E.w w = D)
12	10, 11	anim12i 549	. . . . . . 7 |- ((C e. R /\ D e. S) -> (E.z z = C /\ E.w w = D))
13		eeanv 1913	. . . . . . 7 |- (E.zE.w(z = C /\ w = D) <-> (E.z z = C /\ E.w w = D))
14	12, 13	sylibr 203	. . . . . 6 |- ((C e. R /\ D e. S) -> E.zE.w(z = C /\ w = D))
15	9, 14	anim12i 549	. . . . 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)))
16		ee4anv 1915	. . . . 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)))
17	15, 16	sylibr 203	. . . 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)))
18		cgsex4g.1	. . . . . 6 |- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> ch)
19	18	2eximi 1577	. . . . 5 |- (E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.zE.wch)
20	19	2eximi 1577	. . . 4 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D)) -> E.xE.yE.zE.wch)
21	17, 20	syl 15	. . 3 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> E.xE.yE.zE.wch)
22	1	biimprcd 216	. . . . . 6 |- (ps -> (ch -> ph))
23	22	ancld 536	. . . . 5 |- (ps -> (ch -> (ch /\ ph)))
24	23	2eximdv 1624	. . . 4 |- (ps -> (E.zE.wch -> E.zE.w(ch /\ ph)))
25	24	2eximdv 1624	. . 3 |- (ps -> (E.xE.yE.zE.wch -> E.xE.yE.zE.w(ch /\ ph)))
26	21, 25	syl5com 26	. 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 195	1 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w(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:  copsex4g  4611