Theorem hbsb 2441

Description: If z is not free in ph, it is not free in [ y / x ] ph when y and z are distinct. (Contributed by NM, 12-Aug-1993.)

Hypothesis

Ref	Expression
hbsb.1	|- ( ph -> A. z ph )

Assertion

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

Distinct variable group:   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem hbsb

Step	Hyp	Ref	Expression
1		hbsb.1	. . . 4 |- ( ph -> A. z ph )
2	1	nf5i 2024	. . 3 |- F/ z ph
3	2	nfsb 2440	. 2 |- F/ z [ y / x ] ph
4	3	nf5ri 2065	1 |- ( [ y / x ] ph -> A. z [ 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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  hbab  2613  hblem  2731  bj-hblem  32849