Theorem eqvincf 3331

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 3330	. 2 |- ( A = B <-> E. y ( y = A /\ y = B ) )
3		eqvincf.1	. . . . 5 |- F/_ x A
4	3	nfeq2 2780	. . . 4 |- F/ x y = A
5		eqvincf.2	. . . . 5 |- F/_ x B
6	5	nfeq2 2780	. . . 4 |- F/ x y = B
7	4, 6	nfan 1828	. . 3 |- F/ x ( y = A /\ y = B )
8		nfv 1843	. . 3 |- F/ y ( x = A /\ x = B )
9		eqeq1 2626	. . . 4 |- ( y = x -> ( y = A <-> x = A ) )
10		eqeq1 2626	. . . 4 |- ( y = x -> ( y = B <-> x = B ) )
11	9, 10	anbi12d 747	. . 3 |- ( y = x -> ( ( y = A /\ y = B ) <-> ( x = A /\ x = B ) ) )
12	7, 8, 11	cbvex 2272	. 2 |- ( E. y ( y = A /\ y = B ) <-> E. x ( x = A /\ x = B ) )
13	2, 12	bitri 264	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by: (None)