Theorem nfeqd 2297

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 2134	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1509	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2296	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2296	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1568	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1734	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1452	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   = wceq 1332   F/wnf 1437   e. wcel 1481   F/_wnfc 2269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-nfc 2271

This theorem is referenced by:  nfeld  2298  nfned  2403  vtoclgft  2739  sbcralt  2989  sbcrext  2990  csbiebt  3044  dfnfc2  3762  eusvnfb  4383  eusv2i  4384  iota2df  5120  riota5f  5762