Theorem nfcxfrd 2372

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfceqi.1	|- A = B
nfcxfrd.2	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfcxfrd	|- ( ph -> F/_ x A )

Proof of Theorem nfcxfrd

Step	Hyp	Ref	Expression
1		nfcxfrd.2	. 2 |- ( ph -> F/_ x B )
2		nfceqi.1	. . 3 |- A = B
3	2	nfceqi 2370	. 2 |- ( F/_ x A <-> F/_ x B )
4	1, 3	sylibr 134	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   = wceq 1397   F/_wnfc 2361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-cleq 2224  df-clel 2227  df-nfc 2363

This theorem is referenced by:  nfcsb1d  3158  nfcsbd  3163  nfcsbw  3164  nfifd  3633  nfunid  3900  nfiotadw  5289  nfriotadxy  5979  nfovd  6046  nfnegd  8374