Theorem eqvinc 2741

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 2628	. . . 4 |- E. x x = A
3		ax-1 5	. . . . . 6 |- ( x = A -> ( A = B -> x = A ) )
4		eqtr 2106	. . . . . . 7 |- ( ( x = A /\ A = B ) -> x = B )
5	4	ex 114	. . . . . 6 |- ( x = A -> ( A = B -> x = B ) )
6	3, 5	jca 301	. . . . 5 |- ( x = A -> ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
7	6	eximi 1537	. . . 4 |- ( E. x x = A -> E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
8		pm3.43 570	. . . . 5 |- ( ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> ( A = B -> ( x = A /\ x = B ) ) )
9	8	eximi 1537	. . . 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 1611	. 2 |- ( A = B -> E. x ( x = A /\ x = B ) )
12		eqtr2 2107	. . 3 |- ( ( x = A /\ x = B ) -> A = B )
13	12	exlimiv 1535	. 2 |- ( E. x ( x = A /\ x = B ) -> A = B )
14	11, 13	impbii 125	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   E.wex 1427   e. wcel 1439   _Vcvv 2620

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-v 2622

This theorem is referenced by:  eqvincf  2743