Theorem copsex4g 1904

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 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   x,R,y,z,w   x,S,y,z,w

Proof of Theorem copsex4g

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 |- x e. V
2		visset 1350	. . . . . . . 8 |- y e. V
3	1, 2	opthg 1899	. . . . . . 7 |- (B e. S -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		cleqcom 1103	. . . . . . 7 |- (<.A, B>. = <.x, y>. <-> <.x, y>. = <.A, B>.)
5	3, 4	syl5bb 410	. . . . . 6 |- (B e. S -> (<.A, B>. = <.x, y>. <-> (x = A /\ y = B)))
6	5	adantl 305	. . . . 5 |- ((A e. R /\ B e. S) -> (<.A, B>. = <.x, y>. <-> (x = A /\ y = B)))
7		visset 1350	. . . . . . . 8 |- z e. V
8		visset 1350	. . . . . . . 8 |- w e. V
9	7, 8	opthg 1899	. . . . . . 7 |- (D e. S -> (<.z, w>. = <.C, D>. <-> (z = C /\ w = D)))
10		cleqcom 1103	. . . . . . 7 |- (<.C, D>. = <.z, w>. <-> <.z, w>. = <.C, D>.)
11	9, 10	syl5bb 410	. . . . . 6 |- (D e. S -> (<.C, D>. = <.z, w>. <-> (z = C /\ w = D)))
12	11	adantl 305	. . . . 5 |- ((C e. R /\ D e. S) -> (<.C, D>. = <.z, w>. <-> (z = C /\ w = D)))
13	6, 12	bi2anan9 478	. . . 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 469	. . 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	bi4exdv 940	. 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 9	. . 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 1369	. 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 406	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 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.cop 1810

This theorem is referenced by:  opbrop 2472  oprabval3 3052

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-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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