Theorem nfeqd 2387

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 2223	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1574	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2386	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2386	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1634	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1806	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1521	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   F/_wnfc 2359

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 1493  ax-7 1494  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-cleq 2222  df-nfc 2361

This theorem is referenced by:  nfeld  2388  nfned  2494  vtoclgft  2851  sbcralt  3105  sbcrext  3106  csbiebt  3164  dfnfc2  3906  eusvnfb  4545  eusv2i  4546  iota2df  5304  riotaeqimp  5979  riota5f  5981