Theorem nfcr 2170

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 2167	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
2		sp 1401	. 2 |- ( A. y F/ x y e. A -> F/ x y e. A )
3	1, 2	sylbi 114	1 |- ( F/_ x A -> F/ x y e. 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-4 1400

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

This theorem is referenced by:  nfcrii  2171  nfcrd  2191  abidnf  2709  csbtt  2862  csbnestgf  2898