Theorem eqvincf 2669

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 2667	. 2 |- ( A = B <-> E. y ( y = A /\ y = B ) )
3		eqvincf.1	. . . . 5 |- F/_ x A
4	3	nfeq2 2189	. . . 4 |- F/ x y = A
5		eqvincf.2	. . . . 5 |- F/_ x B
6	5	nfeq2 2189	. . . 4 |- F/ x y = B
7	4, 6	nfan 1457	. . 3 |- F/ x ( y = A /\ y = B )
8		nfv 1421	. . 3 |- F/ y ( x = A /\ x = B )
9		eqeq1 2046	. . . 4 |- ( y = x -> ( y = A <-> x = A ) )
10		eqeq1 2046	. . . 4 |- ( y = x -> ( y = B <-> x = B ) )
11	9, 10	anbi12d 442	. . 3 |- ( y = x -> ( ( y = A /\ y = B ) <-> ( x = A /\ x = B ) ) )
12	7, 8, 11	cbvex 1639	. 2 |- ( E. y ( y = A /\ y = B ) <-> E. x ( x = A /\ x = B ) )
13	2, 12	bitri 173	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   F/_wnfc 2165   _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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by: (None)