Theorem eqvinop 4228

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 4225	. . . . . . 7 |- ( <. x , y >. = <. B , C >. <-> ( x = B /\ y = C ) )
4	3	anbi2i 454	. . . . . 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 1598	. . . 4 |- ( E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) )
9		19.42v 1899	. . . 4 |- ( E. y ( x = B /\ ( y = C /\ A = <. x , y >. ) ) <-> ( x = B /\ E. y ( y = C /\ A = <. x , y >. ) ) )
10		opeq2 3766	. . . . . . 7 |- ( y = C -> <. x , y >. = <. x , C >. )
11	10	eqeq2d 2182	. . . . . 6 |- ( y = C -> ( A = <. x , y >. <-> A = <. x , C >. ) )
12	2, 11	ceqsexv 2769	. . . . 5 |- ( E. y ( y = C /\ A = <. x , y >. ) <-> A = <. x , C >. )
13	12	anbi2i 454	. . . 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 1598	. 2 |- ( E. x E. y ( A = <. x , y >. /\ <. x , y >. = <. B , C >. ) <-> E. x ( x = B /\ A = <. x , C >. ) )
16		opeq1 3765	. . . 4 |- ( x = B -> <. x , C >. = <. B , C >. )
17	16	eqeq2d 2182	. . 3 |- ( x = B -> ( A = <. x , C >. <-> A = <. B , C >. ) )
18	1, 17	ceqsexv 2769	. 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 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   <.cop 3586

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  copsexg  4229  ralxpf  4757  rexxpf  4758  oprabid  5885