Theorem nfcd 2759

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 2082	. 2 |- ( ph -> A. y F/ x y e. A )
4		df-nfc 2753	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
5	3, 4	sylibr 224	1 |- ( ph -> F/_ x A )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-nfc 2753

This theorem is referenced by:  nfabd2  2784  dvelimdc  2786  sbnfc2  4007