Theorem nfs1 1730

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 1452	. . 3 |- ( ph -> A. y ph )
3	2	hbsb3 1729	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
4	3	nfi 1391	1 |- F/ x [ y / x ] ph

Syntax hints:   F/wnf 1389   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-11 1437  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sb8  1777  sb8e  1778