Theorem copsex4g 3540

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 2295	. . . . . . . 8 |- x e. _V
2		visset 2295	. . . . . . . 8 |- y e. _V
3	1, 2	opthg 3533	. . . . . . 7 |- (B e. S -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		eqcom 1886	. . . . . . 7 |- (<.A, B>. = <.x, y>. <-> <.x, y>. = <.A, B>.)
5	3, 4	syl5bb 591	. . . . . 6 |- (B e. S -> (<.A, B>. = <.x, y>. <-> (x = A /\ y = B)))
6	5	adantl 424	. . . . 5 |- ((A e. R /\ B e. S) -> (<.A, B>. = <.x, y>. <-> (x = A /\ y = B)))
7		visset 2295	. . . . . . . 8 |- z e. _V
8		visset 2295	. . . . . . . 8 |- w e. _V
9	7, 8	opthg 3533	. . . . . . 7 |- (D e. S -> (<.z, w>. = <.C, D>. <-> (z = C /\ w = D)))
10		eqcom 1886	. . . . . . 7 |- (<.C, D>. = <.z, w>. <-> <.z, w>. = <.C, D>.)
11	9, 10	syl5bb 591	. . . . . 6 |- (D e. S -> (<.C, D>. = <.z, w>. <-> (z = C /\ w = D)))
12	11	adantl 424	. . . . 5 |- ((C e. R /\ D e. S) -> (<.C, D>. = <.z, w>. <-> (z = C /\ w = D)))
13	6, 12	bi2anan9 694	. . . 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 679	. . 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 1661	. 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 73	. . 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 2323	. 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 587	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  <.cop 3046

This theorem is referenced by:  opbrop 4064  oprabval3 4959

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053

Copyright terms: Public domain