Theorem nfeqd 2192

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 2034	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1421	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2191	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2191	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1480	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1643	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1364	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-nfc 2167

This theorem is referenced by:  nfeld  2193  nfned  2298  vtoclgft  2604  sbcralt  2834  sbcrext  2835  csbiebt  2886  dfnfc2  3598  eusvnfb  4186  eusv2i  4187  iota2df  4891  riota5f  5492