Theorem copsex4g 4002

Description: An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)

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. x E. y E. z E. 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

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

Proof of Theorem copsex4g

Step	Hyp	Ref	Expression
1		eqcom 2083	. . . . . . 7 |- ( <. A , B >. = <. x , y >. <-> <. x , y >. = <. A , B >. )
2		vex 2604	. . . . . . . 8 |- x e. _V
3		vex 2604	. . . . . . . 8 |- y e. _V
4	2, 3	opth 3992	. . . . . . 7 |- ( <. x , y >. = <. A , B >. <-> ( x = A /\ y = B ) )
5	1, 4	bitri 182	. . . . . 6 |- ( <. A , B >. = <. x , y >. <-> ( x = A /\ y = B ) )
6		eqcom 2083	. . . . . . 7 |- ( <. C , D >. = <. z , w >. <-> <. z , w >. = <. C , D >. )
7		vex 2604	. . . . . . . 8 |- z e. _V
8		vex 2604	. . . . . . . 8 |- w e. _V
9	7, 8	opth 3992	. . . . . . 7 |- ( <. z , w >. = <. C , D >. <-> ( z = C /\ w = D ) )
10	6, 9	bitri 182	. . . . . 6 |- ( <. C , D >. = <. z , w >. <-> ( z = C /\ w = D ) )
11	5, 10	anbi12i 447	. . . . 5 |- ( ( <. A , B >. = <. x , y >. /\ <. C , D >. = <. z , w >. ) <-> ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
12	11	anbi1i 445	. . . 4 |- ( ( ( <. A , B >. = <. x , y >. /\ <. C , D >. = <. z , w >. ) /\ ph ) <-> ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ph ) )
13	12	a1i 9	. . 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 ) ) )
14	13	4exbidv 1791	. 2 |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ( <. A , B >. = <. x , y >. /\ <. C , D >. = <. z , w >. ) /\ ph ) <-> E. x E. y E. z E. w ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ph ) ) )
15		id 19	. . 3 |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
16		copsex4g.1	. . 3 |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ( ph <-> ps ) )
17	15, 16	cgsex4g 2636	. 2 |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ph ) <-> ps ) )
18	14, 17	bitrd 186	1 |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ( <. A , B >. = <. x , y >. /\ <. C , D >. = <. z , w >. ) /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by:  opbrop  4437  ovi3  5657  dfplpq2  6544  dfmpq2  6545  enq0breq  6626