Theorem nfcxfrd 2173

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/_xB)

Assertion

Ref	Expression
nfcxfrd	|- (ph -> F/_xA)

Proof of Theorem nfcxfrd

Step	Hyp	Ref	Expression
1		nfcxfrd.2	. 2 |- (ph -> F/_xB)
2		nfceqi.1	. . 3 |- A = B
3	2	nfceqi 2171	. 2 |- ( F/_xA <-> F/_xB)
4	1, 3	sylibr 137	1 |- (ph -> F/_xA)

Syntax hints:   -> wi 4   = wceq 1242   F/_wnfc 2162

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164

This theorem is referenced by:  nfcsb1d  2874  nfcsbd  2877  nfifd  3349  nfunid  3578  nfiotadxy  4813  nfriotadxy  5419  nfovd  5477  nfnegd  6984