Theorem copsex4g 2784

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 1804	. . . . . . . 8 |- x e. V
2		visset 1804	. . . . . . . 8 |- y e. V
3	1, 2	opthg 2778	. . . . . . 7 |- (B e. S -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		eqcom 1469	. . . . . . 7 |- (<.A, B>. = <.x, y>. <-> <.x, y>. = <.A, B>.)
5	3, 4	syl5bb 530	. . . . . 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 1804	. . . . . . . 8 |- z e. V
8		visset 1804	. . . . . . . 8 |- w e. V
9	7, 8	opthg 2778	. . . . . . 7 |- (D e. S -> (<.z, w>. = <.C, D>. <-> (z = C /\ w = D)))
10		eqcom 1469	. . . . . . 7 |- (<.C, D>. = <.z, w>. <-> <.z, w>. = <.C, D>.)
11	9, 10	syl5bb 530	. . . . . 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 630	. . . 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 615	. . 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 1278	. 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 1824	. 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 526	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 953   e. wcel 955  E.wex 977  <.cop 2401

This theorem is referenced by:  opbrop 3228  oprabval3 4015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406

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