Theorem eqvinop 4221

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 4218	. . . . . . 7 |- ( <. x , y >. = <. B , C >. <-> ( x = B /\ y = C ) )
4	3	anbi2i 453	. . . . . 6 |- ( ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( A = <. x , y >. /\ ( x = B /\ y = C ) ) )
5		ancom 264	. . . . . 6 |- ( ( A = <. x , y >. /\ ( x = B /\ y = C ) ) <-> ( ( x = B /\ y = C ) /\ A = <. x , y >. ) )
6		anass 399	. . . . . 6 |- ( ( ( x = B /\ y = C ) /\ A = <. x , y >. ) <-> ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
7	4, 5, 6	3bitri 205	. . . . 5 |- ( ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
8	7	exbii 1593	. . . 4 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
9		19.42v 1894	. . . 4 |- ( E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) )
10		opeq2 3759	. . . . . . 7 |- ( y = C -> <. x , y >. = <. x , C >. )
11	10	eqeq2d 2177	. . . . . 6 |- ( y = C -> ( A = <. x , y >. <-> A = <. x , C >. ) )
12	2, 11	ceqsexv 2765	. . . . 5 |- ( E. y ( y = C /\ A = <. x , y >. ) <-> A = <. x , C >. )
13	12	anbi2i 453	. . . 4 |- ( ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ A = <. x , C >. ) )
14	8, 9, 13	3bitri 205	. . 3 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( x = B /\ A = <. x , C >. ) )
15	14	exbii 1593	. 2 |- ( E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. x ( x = B /\ A = <. x , C >. ) )
16		opeq1 3758	. . . 4 |- ( x = B -> <. x , C >. = <. B , C >. )
17	16	eqeq2d 2177	. . 3 |- ( x = B -> ( A = <. x , C >. <-> A = <. B , C >. ) )
18	1, 17	ceqsexv 2765	. 2 |- ( E. x ( x = B /\ A = <. x , C >. ) <-> A = <. B , C >. )
19	15, 18	bitr2i 184	1 |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   <.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:  copsexg  4222  ralxpf  4750  rexxpf  4751  oprabid  5874