Theorem nfcd 2173

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 1415	. 2 |- ( ph -> A. y F/ x y e. A )
4		df-nfc 2167	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
5	3, 4	sylibr 137	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   A.wal 1241   F/wnf 1349   e. wcel 1393   F/_wnfc 2165

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 1336  ax-gen 1338  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-nfc 2167

This theorem is referenced by:  nfabd  2196  dvelimdc  2197  sbnfc2  2906