Theorem eqvinc 2667

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 2563	. . . 4 |- E. x x = A
3		ax-1 5	. . . . . 6 |- ( x = A -> ( A = B -> x = A ) )
4		eqtr 2057	. . . . . . 7 |- ( ( x = A /\ A = B ) -> x = B )
5	4	ex 108	. . . . . 6 |- ( x = A -> ( A = B -> x = B ) )
6	3, 5	jca 290	. . . . 5 |- ( x = A -> ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
7	6	eximi 1491	. . . 4 |- ( E. x x = A -> E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
8		pm3.43 534	. . . . 5 |- ( ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> ( A = B -> ( x = A /\ x = B ) ) )
9	8	eximi 1491	. . . 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 1565	. 2 |- ( A = B -> E. x ( x = A /\ x = B ) )
12		eqtr2 2058	. . 3 |- ( ( x = A /\ x = B ) -> A = B )
13	12	exlimiv 1489	. 2 |- ( E. x ( x = A /\ x = B ) -> A = B )
14	11, 13	impbii 117	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  eqvincf  2669