Theorem nfnf 1845

Description: If x is not free in ph, it is not free in F/yph. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
nfal.1	|- F/xph

Assertion

Ref	Expression
nfnf	|- F/x F/yph

Proof of Theorem nfnf

Step	Hyp	Ref	Expression
1		df-nf 1545	. 2 |- ( F/yph <-> A.y(ph -> A.yph))
2		nfal.1	. . . 4 |- F/xph
3	2	nfal 1842	. . . 4 |- F/xA.yph
4	2, 3	nfim 1813	. . 3 |- F/x(ph -> A.yph)
5	4	nfal 1842	. 2 |- F/xA.y(ph -> A.yph)
6	1, 5	nfxfr 1570	1 |- F/x F/yph

Syntax hints:   -> wi 4  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfnfc  2496