Theorem hbsb3 1796

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 1752	. 2 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
3		hbsb2a 1794	. 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 1341   [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-11 1494  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-sb 1751

This theorem is referenced by:  nfs1  1797  sbcof2  1798  ax16  1801  sb8h  1842  sb8eh  1843  ax16ALT  1847