Theorem nfeqd 2243

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 2082	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1466	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2242	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2242	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1525	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1690	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1409	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   = wceq 1289   F/wnf 1394   e. wcel 1438   F/_wnfc 2215

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 1381  ax-7 1382  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-nfc 2217

This theorem is referenced by:  nfeld  2244  nfned  2349  vtoclgft  2669  sbcralt  2915  sbcrext  2916  csbiebt  2967  dfnfc2  3671  eusvnfb  4276  eusv2i  4277  iota2df  5004  riota5f  5632