Theorem nfcr 2300

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Assertion

Ref	Expression
nfcr	|- ( F/_ x A -> F/ x y e. A )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem nfcr

Step	Hyp	Ref	Expression
1		df-nfc 2297	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
2		sp 1499	. 2 |- ( A. y F/ x y e. A -> F/ x y e. A )
3	1, 2	sylbi 120	1 |- ( F/_ x A -> F/ x y e. A )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448   e. wcel 2136   F/_wnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nfc 2297

This theorem is referenced by:  nfcrii  2301  nfcrd  2322  abidnf  2894  csbtt  3057  csbnestgf  3097