Theorem hbsb 2000

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 1508	. . 3 |- F/ z ph
3	2	nfsb 1997	. 2 |- F/ z [ y / x ] ph
4	3	nfri 1565	1 |- ( [ y / x ] ph -> A. z [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1393   [wsb 1808

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sb10f  2046  hbsb4  2063  sb8euh  2100  hbab  2220  hblem  2337