Theorem eqvinop 4160

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 4157	. . . . . . 7 |- ( <. x , y >. = <. B , C >. <-> ( x = B /\ y = C ) )
4	3	anbi2i 452	. . . . . 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 398	. . . . . 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 1584	. . . 4 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
9		19.42v 1878	. . . 4 |- ( E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) )
10		opeq2 3701	. . . . . . 7 |- ( y = C -> <. x , y >. = <. x , C >. )
11	10	eqeq2d 2149	. . . . . 6 |- ( y = C -> ( A = <. x , y >. <-> A = <. x , C >. ) )
12	2, 11	ceqsexv 2720	. . . . 5 |- ( E. y ( y = C /\ A = <. x , y >. ) <-> A = <. x , C >. )
13	12	anbi2i 452	. . . 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 1584	. 2 |- ( E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. x ( x = B /\ A = <. x , C >. ) )
16		opeq1 3700	. . . 4 |- ( x = B -> <. x , C >. = <. B , C >. )
17	16	eqeq2d 2149	. . 3 |- ( x = B -> ( A = <. x , C >. <-> A = <. B , C >. ) )
18	1, 17	ceqsexv 2720	. 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 1331   E.wex 1468   e. wcel 1480   _Vcvv 2681   <.cop 3525

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531

This theorem is referenced by:  copsexg  4161  ralxpf  4680  rexxpf  4681  oprabid  5796