Theorem hbsb 1864

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 1391	. . 3 |- F/ z ph
3	2	nfsb 1863	. 2 |- F/ z [ y / x ] ph
4	3	nfri 1452	1 |- ( [ y / x ] ph -> A. z [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sb10f  1912  hbsb4  1929  sb8euh  1964  hbab  2072  hblem  2186