Theorem nfcxfrd 2317

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 2315	. 2 |- ( F/_ x A <-> F/_ x B )
4	1, 3	sylibr 134	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   = wceq 1353   F/_wnfc 2306

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by:  nfcsb1d  3088  nfcsbd  3092  nfcsbw  3093  nfifd  3561  nfunid  3816  nfiotadw  5181  nfriotadxy  5838  nfovd  5903  nfnegd  8151