Theorem nfs1 2044

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/yph

Assertion

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

Proof of Theorem nfs1

Step	Hyp	Ref	Expression
1		nfs1.1	. . . 4 |- F/yph
2	1	nfri 1762	. . 3 |- (ph -> A.yph)
3	2	hbsb3 2043	. 2 |- ([y / x]ph -> A.x[y / x]ph)
4	3	nfi 1551	1 |- F/x[y / x]ph

Syntax hints:   F/wnf 1544  [wsb 1648

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  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  ax16ALT2  2048  sbco2  2086  sb8  2092  mo  2226  ralxpf  4827