Theorem nfeqd 2327

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 2164	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1521	. . 3 |- F/ y ph
3		nfeqd.1	. . . . 5 |- ( ph -> F/_ x A )
4	3	nfcrd 2326	. . . 4 |- ( ph -> F/ x y e. A )
5		nfeqd.2	. . . . 5 |- ( ph -> F/_ x B )
6	5	nfcrd 2326	. . . 4 |- ( ph -> F/ x y e. B )
7	4, 6	nfbid 1581	. . 3 |- ( ph -> F/ x ( y e. A <-> y e. B ) )
8	2, 7	nfald 1753	. 2 |- ( ph -> F/ x A. y ( y e. A <-> y e. B ) )
9	1, 8	nfxfrd 1468	1 |- ( ph -> F/ x A = B )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   F/wnf 1453   e. wcel 2141   F/_wnfc 2299

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 1440  ax-7 1441  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-nfc 2301

This theorem is referenced by:  nfeld  2328  nfned  2434  vtoclgft  2780  sbcralt  3031  sbcrext  3032  csbiebt  3088  dfnfc2  3814  eusvnfb  4439  eusv2i  4440  iota2df  5184  riota5f  5833