Theorem hbsb 2009

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.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

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	nfi 1515	. . 3 |- F/ z ph
3	2	nfsb 2006	. 2 |- F/ z [ y / x ] ph
4	3	nfri 1572	1 |- ( [ y / x ] ph -> A. z [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1400   [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sb10f  2055  hbsb4  2072  sb8euh  2109  hbab  2229  hblem  2346