Theorem eqvincf 2967

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/_xA
eqvincf.2	|- F/_xB
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 2966	. 2 |- (A = B <-> E.y(y = A /\ y = B))
3		eqvincf.1	. . . . 5 |- F/_xA
4	3	nfeq2 2500	. . . 4 |- F/x y = A
5		eqvincf.2	. . . . 5 |- F/_xB
6	5	nfeq2 2500	. . . 4 |- F/x y = B
7	4, 6	nfan 1824	. . 3 |- F/x(y = A /\ y = B)
8		nfv 1619	. . 3 |- F/y(x = A /\ x = B)
9		eqeq1 2359	. . . 4 |- (y = x -> (y = A <-> x = A))
10		eqeq1 2359	. . . 4 |- (y = x -> (y = B <-> x = B))
11	9, 10	anbi12d 691	. . 3 |- (y = x -> ((y = A /\ y = B) <-> (x = A /\ x = B)))
12	7, 8, 11	cbvex 1985	. 2 |- (E.y(y = A /\ y = B) <-> E.x(x = A /\ x = B))
13	2, 12	bitri 240	1 |- (A = B <-> E.x(x = A /\ x = B))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   F/_wnfc 2476  _Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by: (None)