Theorem eqvincf 2905

Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)

Hypotheses

Ref	Expression
eqvincf.1	|- F/_ x A
eqvincf.2	|- F/_ x B
eqvincf.3	|- A e. _V

Assertion

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

Proof of Theorem eqvincf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvincf.3	. . 3 |- A e. _V
2	1	eqvinc 2903	. 2 |- ( A = B <-> E. y ( y = A /\ y = B ) )
3		eqvincf.1	. . . . 5 |- F/_ x A
4	3	nfeq2 2362	. . . 4 |- F/ x y = A
5		eqvincf.2	. . . . 5 |- F/_ x B
6	5	nfeq2 2362	. . . 4 |- F/ x y = B
7	4, 6	nfan 1589	. . 3 |- F/ x ( y = A /\ y = B )
8		nfv 1552	. . 3 |- F/ y ( x = A /\ x = B )
9		eqeq1 2214	. . . 4 |- ( y = x -> ( y = A <-> x = A ) )
10		eqeq1 2214	. . . 4 |- ( y = x -> ( y = B <-> x = B ) )
11	9, 10	anbi12d 473	. . 3 |- ( y = x -> ( ( y = A /\ y = B ) <-> ( x = A /\ x = B ) ) )
12	7, 8, 11	cbvex 1780	. 2 |- ( E. y ( y = A /\ y = B ) <-> E. x ( x = A /\ x = B ) )
13	2, 12	bitri 184	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   F/_wnfc 2337   _Vcvv 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by: (None)