Theorem eqvinop 1901

Description: A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109.

Hypotheses

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

Assertion

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

Distinct variable group(s):   x,y,A   x,B,y   x,C,y

Proof of Theorem eqvinop

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 |- x e. V
2		visset 1350	. . . . . . . 8 |- y e. V
3		eqvinop.2	. . . . . . . 8 |- C e. V
4	1, 2, 3	opth 1898	. . . . . . 7 |- (<.x, y>. = <.B, C>. <-> (x = B /\ y = C))
5	4	anbi2i 367	. . . . . 6 |- ((A = <.x, y>. /\ <.x, y>. = <.B, C>.) <-> (A = <.x, y>. /\ (x = B /\ y = C)))
6		ancom 333	. . . . . 6 |- ((A = <.x, y>. /\ (x = B /\ y = C)) <-> ((x = B /\ y = C) /\ A = <.x, y>.))
7		anass 336	. . . . . 6 |- (((x = B /\ y = C) /\ A = <.x, y>.) <-> (x = B /\ (y = C /\ A = <.x, y>.)))
8	5, 6, 7	3bitr 155	. . . . 5 |- ((A = <.x, y>. /\ <.x, y>. = <.B, C>.) <-> (x = B /\ (y = C /\ A = <.x, y>.)))
9	8	biex 733	. . . 4 |- (E.y(A = <.x, y>. /\ <.x, y>. = <.B, C>.) <-> E.y(x = B /\ (y = C /\ A = <.x, y>.)))
10		19.42v 966	. . . 4 |- (E.y(x = B /\ (y = C /\ A = <.x, y>.)) <-> (x = B /\ E.y(y = C /\ A = <.x, y>.)))
11		opeq2 1877	. . . . . . 7 |- (y = C -> <.x, y>. = <.x, C>.)
12	11	cleq2d 1112	. . . . . 6 |- (y = C -> (A = <.x, y>. <-> A = <.x, C>.))
13	3, 12	ceqsexv 1371	. . . . 5 |- (E.y(y = C /\ A = <.x, y>.) <-> A = <.x, C>.)
14	13	anbi2i 367	. . . 4 |- ((x = B /\ E.y(y = C /\ A = <.x, y>.)) <-> (x = B /\ A = <.x, C>.))
15	9, 10, 14	3bitr 155	. . 3 |- (E.y(A = <.x, y>. /\ <.x, y>. = <.B, C>.) <-> (x = B /\ A = <.x, C>.))
16	15	biex 733	. 2 |- (E.xE.y(A = <.x, y>. /\ <.x, y>. = <.B, C>.) <-> E.x(x = B /\ A = <.x, C>.))
17		eqvinop.1	. . 3 |- B e. V
18		opeq1 1876	. . . 4 |- (x = B -> <.x, C>. = <.B, C>.)
19	18	cleq2d 1112	. . 3 |- (x = B -> (A = <.x, C>. <-> A = <.B, C>.))
20	17, 19	ceqsexv 1371	. 2 |- (E.x(x = B /\ A = <.x, C>.) <-> A = <.B, C>.)
21	16, 20	bitr2 152	1 |- (A = <.B, C>. <-> E.xE.y(A = <.x, y>. /\ <.x, y>. = <.B, C>.))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810

This theorem is referenced by:  copsexg 1902

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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