Theorem nfeqd 2244

Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfeqd.1	|- ( ph -> F/_ x A )
nfeqd.2	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfeqd	|- ( ph -> F/ x A = B )

Proof of Theorem nfeqd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2083	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1467	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2243	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2243	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1526	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1691	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1410	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1288   = wceq 1290   F/wnf 1395   e. wcel 1439   F/_wnfc 2216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-cleq 2082  df-nfc 2218

This theorem is referenced by:  nfeld  2245  nfned  2350  vtoclgft  2670  sbcralt  2916  sbcrext  2917  csbiebt  2968  dfnfc2  3677  eusvnfb  4289  eusv2i  4290  iota2df  5017  riota5f  5646