Theorem nfcd 2294

Description: Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfcd.1	|- F/ y ph
nfcd.2	|- ( ph -> F/ x y e. A )

Assertion

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

Distinct variable groups:   x, y   y, A

Allowed substitution hints:   ph( x, y)   A( x)

Proof of Theorem nfcd

Step	Hyp	Ref	Expression
1		nfcd.1	. . 3 |- F/ y ph
2		nfcd.2	. . 3 |- ( ph -> F/ x y e. A )
3	1, 2	alrimi 1502	. 2 |- ( ph -> A. y F/ x y e. A )
4		df-nfc 2288	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
5	3, 4	sylibr 133	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   A.wal 1333   F/wnf 1440   e. wcel 2128   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-4 1490

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-nfc 2288

This theorem is referenced by:  nfabdw  2318  nfabd  2319  dvelimdc  2320  sbnfc2  3091