Theorem hbsb 1961

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 1473	. . 3 |- F/ z ph
3	2	nfsb 1958	. 2 |- F/ z [ y / x ] ph
4	3	nfri 1530	1 |- ( [ y / x ] ph -> A. z [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1362   [wsb 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sb10f  2007  hbsb4  2024  sb8euh  2061  hbab  2180  hblem  2297