Theorem nfcxfrd 2306

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 2304	. 2 |- ( F/_ x A <-> F/_ x B )
4	1, 3	sylibr 133	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   = wceq 1343   F/_wnfc 2295

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  nfcsb1d  3076  nfcsbd  3080  nfcsbw  3081  nfifd  3547  nfunid  3796  nfiotadw  5156  nfriotadxy  5806  nfovd  5871  nfnegd  8094