Theorem nfeqd 2365

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 2201	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1552	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2364	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2364	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1612	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1784	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1499	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   e. wcel 2178   F/_wnfc 2337

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 1471  ax-7 1472  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2200  df-nfc 2339

This theorem is referenced by:  nfeld  2366  nfned  2472  vtoclgft  2828  sbcralt  3082  sbcrext  3083  csbiebt  3141  dfnfc2  3882  eusvnfb  4519  eusv2i  4520  iota2df  5276  riota5f  5947