Theorem copsex4g 4225

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 2167	. . . . . . 7 |- ( <. A , B >. = <. x , y >. <-> <. x , y >. = <. A , B >. )
2		vex 2729	. . . . . . . 8 |- x e. _V
3		vex 2729	. . . . . . . 8 |- y e. _V
4	2, 3	opth 4215	. . . . . . 7 |- ( <. x , y >. = <. A , B >. <-> ( x = A /\ y = B ) )
5	1, 4	bitri 183	. . . . . 6 |- ( <. A , B >. = <. x , y >. <-> ( x = A /\ y = B ) )
6		eqcom 2167	. . . . . . 7 |- ( <. C , D >. = <. z , w >. <-> <. z , w >. = <. C , D >. )
7		vex 2729	. . . . . . . 8 |- z e. _V
8		vex 2729	. . . . . . . 8 |- w e. _V
9	7, 8	opth 4215	. . . . . . 7 |- ( <. z , w >. = <. C , D >. <-> ( z = C /\ w = D ) )
10	6, 9	bitri 183	. . . . . 6 |- ( <. C , D >. = <. z , w >. <-> ( z = C /\ w = D ) )
11	5, 10	anbi12i 456	. . . . 5 |- ( ( <. A , B >. = <. x , y >. /\ <. C , D >. = <. z , w >. ) <-> ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
12	11	anbi1i 454	. . . 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 1858	. 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 2763	. 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 187	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 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   <.cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by:  opbrop  4683  ovi3  5978  dfplpq2  7295  dfmpq2  7296  enq0breq  7377