Theorem nfcxfrd 2297

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 2295	. 2 |- ( F/_ x A <-> F/_ x B )
4	1, 3	sylibr 133	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   = wceq 1335   F/_wnfc 2286

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288

This theorem is referenced by:  nfcsb1d  3062  nfcsbd  3066  nfcsbw  3067  nfifd  3532  nfunid  3779  nfiotadw  5137  nfriotadxy  5785  nfovd  5847  nfnegd  8065