Theorem hbsb3 2364

Description: If y is not free in ph, x is not free in [ y / x ] ph. (Contributed by NM, 14-May-1993.)

Hypothesis

Ref	Expression
hbsb3.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
hbsb3	|- ( [ y / x ] ph -> A. x [ y / x ] ph )

Proof of Theorem hbsb3

Step	Hyp	Ref	Expression
1		hbsb3.1	. . 3 |- ( ph -> A. y ph )
2	1	sbimi 1886	. 2 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
3		hbsb2a 2361	. 2 |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )
4	2, 3	syl 17	1 |- ( [ y / x ] ph -> A. x [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  nfs1  2365  axc16ALT  2366