Theorem nfci 2245

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

Hypothesis

Ref	Expression
nfci.1	|- F/ x y e. A

Assertion

Ref	Expression
nfci	|- F/_ x A

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem nfci

Step	Hyp	Ref	Expression
1		df-nfc 2244	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
2		nfci.1	. 2 |- F/ x y e. A
3	1, 2	mpgbir 1412	1 |- F/_ x A

Syntax hints:   F/wnf 1419   e. wcel 1463   F/_wnfc 2242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1408

This theorem depends on definitions:  df-bi 116  df-nfc 2244

This theorem is referenced by:  nfcii  2246  nfcv  2255  nfab1  2257  nfab  2260