Theorem hbsb3 1822

Description: If y is not free in ph, x is not free in [ y / x ] ph. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbsb3.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
hbsb3	|- ( [ y / x ] ph -> A. x [ y / x ] ph )

Proof of Theorem hbsb3

Step	Hyp	Ref	Expression
1		hbsb3.1	. . 3 |- ( ph -> A. y ph )
2	1	sbimi 1778	. 2 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
3		hbsb2a 1820	. 2 |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )
4	2, 3	syl 14	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-11 1520  ax-4 1524  ax-i9 1544  ax-ial 1548

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

This theorem is referenced by:  nfs1  1823  sbcof2  1824  ax16  1827  sb8h  1868  sb8eh  1869  ax16ALT  1873