Theorem nfs1 1857

Description: If y is not free in ph, x is not free in [ y / x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfs1.1	|- F/ y ph

Assertion

Ref	Expression
nfs1	|- F/ x [ y / x ] ph

Proof of Theorem nfs1

Step	Hyp	Ref	Expression
1		nfs1.1	. . . 4 |- F/ y ph
2	1	nfri 1567	. . 3 |- ( ph -> A. y ph )
3	2	hbsb3 1856	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
4	3	nfi 1510	1 |- F/ x [ y / x ] ph

Syntax hints:   F/wnf 1508   [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-11 1554  ax-4 1558  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sb8  1904  sb8e  1905