Theorem eqvinc 2740

Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
eqvinc.1	|- A e. _V

Assertion

Ref	Expression
eqvinc	|- ( A = B <-> E. x ( x = A /\ x = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem eqvinc

Step	Hyp	Ref	Expression
1		eqvinc.1	. . . . 5 |- A e. _V
2	1	isseti 2627	. . . 4 |- E. x x = A
3		ax-1 5	. . . . . 6 |- ( x = A -> ( A = B -> x = A ) )
4		eqtr 2105	. . . . . . 7 |- ( ( x = A /\ A = B ) -> x = B )
5	4	ex 113	. . . . . 6 |- ( x = A -> ( A = B -> x = B ) )
6	3, 5	jca 300	. . . . 5 |- ( x = A -> ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
7	6	eximi 1536	. . . 4 |- ( E. x x = A -> E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
8		pm3.43 569	. . . . 5 |- ( ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> ( A = B -> ( x = A /\ x = B ) ) )
9	8	eximi 1536	. . . 4 |- ( E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> E. x ( A = B -> ( x = A /\ x = B ) ) )
10	2, 7, 9	mp2b 8	. . 3 |- E. x ( A = B -> ( x = A /\ x = B ) )
11	10	19.37aiv 1610	. 2 |- ( A = B -> E. x ( x = A /\ x = B ) )
12		eqtr2 2106	. . 3 |- ( ( x = A /\ x = B ) -> A = B )
13	12	exlimiv 1534	. 2 |- ( E. x ( x = A /\ x = B ) -> A = B )
14	11, 13	impbii 124	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621

This theorem is referenced by:  eqvincf  2742