Theorem copsex4g 4333

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 2231	. . . . . . 7 |- ( <. A , B >. = <. x , y >. <-> <. x , y >. = <. A , B >. )
2		vex 2802	. . . . . . . 8 |- x e. _V
3		vex 2802	. . . . . . . 8 |- y e. _V
4	2, 3	opth 4323	. . . . . . 7 |- ( <. x , y >. = <. A , B >. <-> ( x = A /\ y = B ) )
5	1, 4	bitri 184	. . . . . 6 |- ( <. A , B >. = <. x , y >. <-> ( x = A /\ y = B ) )
6		eqcom 2231	. . . . . . 7 |- ( <. C , D >. = <. z , w >. <-> <. z , w >. = <. C , D >. )
7		vex 2802	. . . . . . . 8 |- z e. _V
8		vex 2802	. . . . . . . 8 |- w e. _V
9	7, 8	opth 4323	. . . . . . 7 |- ( <. z , w >. = <. C , D >. <-> ( z = C /\ w = D ) )
10	6, 9	bitri 184	. . . . . 6 |- ( <. C , D >. = <. z , w >. <-> ( z = C /\ w = D ) )
11	5, 10	anbi12i 460	. . . . 5 |- ( ( <. A , B >. = <. x , y >. /\ <. C , D >. = <. z , w >. ) <-> ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
12	11	anbi1i 458	. . . 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 1916	. 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 2837	. 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 188	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 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   <.cop 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  opbrop  4798  ovi3  6142  dfplpq2  7541  dfmpq2  7542  enq0breq  7623