Theorem nfs1 2194

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 1956	. . 3 |- ( ph -> A. y ph )
3	2	hbsb3 2193	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
4	3	nfi 1677	1 |- F/ x [ y / x ] ph

Syntax hints:   F/wnf 1670   [wsb 1800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-12 1936  ax-13 2091

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1667  df-nf 1671  df-sb 1801

This theorem is referenced by:  sb8  2253  sb8e  2254