Theorem nfeqd 2233

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 2075	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1461	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2232	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2232	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1520	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1683	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1404	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   e. wcel 1433   F/_wnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-nfc 2208

This theorem is referenced by:  nfeld  2234  nfned  2338  vtoclgft  2649  sbcralt  2890  sbcrext  2891  csbiebt  2942  dfnfc2  3619  eusvnfb  4204  eusv2i  4205  iota2df  4911  riota5f  5512