Theorem eqvinop 4258

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 4255	. . . . . . 7 |- ( <. x , y >. = <. B , C >. <-> ( x = B /\ y = C ) )
4	3	anbi2i 457	. . . . . 6 |- ( ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( A = <. x , y >. /\ ( x = B /\ y = C ) ) )
5		ancom 266	. . . . . 6 |- ( ( A = <. x , y >. /\ ( x = B /\ y = C ) ) <-> ( ( x = B /\ y = C ) /\ A = <. x , y >. ) )
6		anass 401	. . . . . 6 |- ( ( ( x = B /\ y = C ) /\ A = <. x , y >. ) <-> ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
7	4, 5, 6	3bitri 206	. . . . 5 |- ( ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
8	7	exbii 1616	. . . 4 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
9		19.42v 1918	. . . 4 |- ( E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) )
10		opeq2 3794	. . . . . . 7 |- ( y = C -> <. x , y >. = <. x , C >. )
11	10	eqeq2d 2201	. . . . . 6 |- ( y = C -> ( A = <. x , y >. <-> A = <. x , C >. ) )
12	2, 11	ceqsexv 2791	. . . . 5 |- ( E. y ( y = C /\ A = <. x , y >. ) <-> A = <. x , C >. )
13	12	anbi2i 457	. . . 4 |- ( ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ A = <. x , C >. ) )
14	8, 9, 13	3bitri 206	. . 3 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> ( x = B /\ A = <. x , C >. ) )
15	14	exbii 1616	. 2 |- ( E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. x ( x = B /\ A = <. x , C >. ) )
16		opeq1 3793	. . . 4 |- ( x = B -> <. x , C >. = <. B , C >. )
17	16	eqeq2d 2201	. . 3 |- ( x = B -> ( A = <. x , C >. <-> A = <. B , C >. ) )
18	1, 17	ceqsexv 2791	. 2 |- ( E. x ( x = B /\ A = <. x , C >. ) <-> A = <. B , C >. )
19	15, 18	bitr2i 185	1 |- ( A = <. B , C >. <-> E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   _Vcvv 2752   <.cop 3610

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616

This theorem is referenced by:  copsexg  4259  ralxpf  4788  rexxpf  4789  oprabid  5923