Theorem eqvinc 2661

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 2557	. . . 4 |- E.x x = A
3		ax-1 5	. . . . . 6 |- (x = A -> (A = B -> x = A))
4		eqtr 2054	. . . . . . 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 1488	. . . 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 1488	. . . 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 1562	. 2 |- (A = B -> E.x(x = A /\ x = B))
12		eqtr2 2055	. . 3 |- ((x = A /\ x = B) -> A = B)
13	12	exlimiv 1486	. 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 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  eqvincf  2663