Theorem eqvinop 3998

Description: A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)

Hypotheses

Ref	Expression
eqvinop.1	|- B e. _V
eqvinop.2	|- C e. _V

Assertion

Ref	Expression
eqvinop	|- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Proof of Theorem eqvinop

Step	Hyp	Ref	Expression
1		eqvinop.1	. . . . . . . 8 |- B e. _V
2		eqvinop.2	. . . . . . . 8 |- C e. _V
3	1, 2	opth2 3995	. . . . . . 7 |- ( <. x , y >. = <. B , C >. <-> ( x = B /\ y = C ) )
4	3	anbi2i 444	. . . . . 6 |- ( ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( A = <. x , y >. /\ ( x = B /\ y = C ) ) )
5		ancom 262	. . . . . 6 |- ( ( A = <. x , y >. /\ ( x = B /\ y = C ) ) <-> ( ( x = B /\ y = C ) /\ A = <. x , y >. ) )
6		anass 393	. . . . . 6 |- ( ( ( x = B /\ y = C ) /\ A = <. x , y >. ) <-> ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
7	4, 5, 6	3bitri 204	. . . . 5 |- ( ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
8	7	exbii 1536	. . . 4 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
9		19.42v 1827	. . . 4 |- ( E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) )
10		opeq2 3571	. . . . . . 7 |- ( y = C -> <. x , y >. = <. x , C >. )
11	10	eqeq2d 2092	. . . . . 6 |- ( y = C -> ( A = <. x , y >. <-> A = <. x , C >. ) )
12	2, 11	ceqsexv 2638	. . . . 5 |- ( E. y ( y = C /\ A = <. x , y >. ) <-> A = <. x , C >. )
13	12	anbi2i 444	. . . 4 |- ( ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ A = <. x , C >. ) )
14	8, 9, 13	3bitri 204	. . 3 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( x = B /\ A = <. x , C >. ) )
15	14	exbii 1536	. 2 |- ( E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. x ( x = B /\ A = <. x , C >. ) )
16		opeq1 3570	. . . 4 |- ( x = B -> <. x , C >. = <. B , C >. )
17	16	eqeq2d 2092	. . 3 |- ( x = B -> ( A = <. x , C >. <-> A = <. B , C >. ) )
18	1, 17	ceqsexv 2638	. 2 |- ( E. x ( x = B /\ A = <. x , C >. ) <-> A = <. B , C >. )
19	15, 18	bitr2i 183	1 |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   <.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:  copsexg  3999  ralxpf  4500  rexxpf  4501  oprabid  5557