Theorem hbs1 1957

Description: x is not free in [ y / x ] ph when x and y are distinct. (Contributed by NM, 5-Aug-1993.) (Proof by Jim Kingdon, 16-Dec-2017.) (New usage is discouraged.)

Assertion

Ref	Expression
hbs1	|- ( [ y / x ] ph -> A. x [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem hbs1

Step	Hyp	Ref	Expression
1		sb6 1901	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		ax-ial 1548	. 2 |- ( A. x ( x = y -> ph ) -> A. x A. x ( x = y -> ph ) )
3	1, 2	hbxfrbi 1486	1 |- ( [ y / x ] ph -> A. x [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1362   [wsb 1776

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-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  nfs1v  1958  sb9v  1997  eu1  2070  mopick  2123  hbab1  2185