Theorem nfeqd 2389

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 2225	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1576	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2388	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2388	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1636	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1808	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1523	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   = wceq 1397   F/wnf 1508   e. wcel 2202   F/_wnfc 2361

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-5 1495  ax-7 1496  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-cleq 2224  df-nfc 2363

This theorem is referenced by:  nfeld  2390  nfned  2496  vtoclgft  2854  sbcralt  3108  sbcrext  3109  csbiebt  3167  dfnfc2  3911  eusvnfb  4551  eusv2i  4552  iota2df  5312  riotaeqimp  5995  riota5f  5997