Theorem copsex4g 2789

Description: An implicit substitution inference for 2 ordered pairs.

Hypothesis

Ref	Expression
copsex4g.1	|- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> (ph <-> ps))

Assertion

Ref	Expression
copsex4g	|- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w((<.A, B>. = <.x, y>. /\ <.C, D>. = <.z, w>.) /\ 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   x,R,y,z,w   x,S,y,z,w

Proof of Theorem copsex4g

Step	Hyp	Ref	Expression
1		visset 1809	. . . . . . . 8 |- x e. V
2		visset 1809	. . . . . . . 8 |- y e. V
3	1, 2	opthg 2783	. . . . . . 7 |- (B e. S -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		eqcom 1474	. . . . . . 7 |- (<.A, B>. = <.x, y>. <-> <.x, y>. = <.A, B>.)
5	3, 4	syl5bb 531	. . . . . 6 |- (B e. S -> (<.A, B>. = <.x, y>. <-> (x = A /\ y = B)))
6	5	adantl 388	. . . . 5 |- ((A e. R /\ B e. S) -> (<.A, B>. = <.x, y>. <-> (x = A /\ y = B)))
7		visset 1809	. . . . . . . 8 |- z e. V
8		visset 1809	. . . . . . . 8 |- w e. V
9	7, 8	opthg 2783	. . . . . . 7 |- (D e. S -> (<.z, w>. = <.C, D>. <-> (z = C /\ w = D)))
10		eqcom 1474	. . . . . . 7 |- (<.C, D>. = <.z, w>. <-> <.z, w>. = <.C, D>.)
11	9, 10	syl5bb 531	. . . . . 6 |- (D e. S -> (<.C, D>. = <.z, w>. <-> (z = C /\ w = D)))
12	11	adantl 388	. . . . 5 |- ((C e. R /\ D e. S) -> (<.C, D>. = <.z, w>. <-> (z = C /\ w = D)))
13	6, 12	bi2anan9 631	. . . 4 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> ((<.A, B>. = <.x, y>. /\ <.C, D>. = <.z, w>.) <-> ((x = A /\ y = B) /\ (z = C /\ w = D))))
14	13	anbi1d 616	. . 3 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (((<.A, B>. = <.x, y>. /\ <.C, D>. = <.z, w>.) /\ ph) <-> (((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ph)))
15	14	4exbidv 1281	. 2 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w((<.A, B>. = <.x, y>. /\ <.C, D>. = <.z, w>.) /\ ph) <-> E.xE.yE.zE.w(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ph)))
16		id 59	. . 3 |- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> ((x = A /\ y = B) /\ (z = C /\ w = D)))
17		copsex4g.1	. . 3 |- (((x = A /\ y = B) /\ (z = C /\ w = D)) -> (ph <-> ps))
18	16, 17	cgsex4g 1829	. 2 |- (((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)) /\ ph) <-> ps))
19	15, 18	bitrd 527	1 |- (((A e. R /\ B e. S) /\ (C e. R /\ D e. S)) -> (E.xE.yE.zE.w((<.A, B>. = <.x, y>. /\ <.C, D>. = <.z, w>.) /\ ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  <.cop 2407

This theorem is referenced by:  opbrop 3233  oprabval3 4021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

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