Theorem nfeqd 2407

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 2232	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1581	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2406	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2406	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1641	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1813	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1528	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   F/wnf 1513   e. wcel 2209   F/_wnfc 2379

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 1500  ax-7 1501  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-nfc 2381

This theorem is referenced by:  nfeld  2408  nfned  2514  vtoclgft  2873  sbcralt  3128  sbcrext  3129  csbiebt  3187  dfnfc2  3948  eusvnfb  4595  eusv2i  4596  iota2df  5358  riotaeqimp  6053  riota5f  6055