Theorem sbrim 1944

Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sbrim.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
sbrim	|- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )

Proof of Theorem sbrim

Step	Hyp	Ref	Expression
1		sbim 1941	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbrim.1	. . . 4 |- ( ph -> A. x ph )
3	2	sbh 1764	. . 3 |- ( [ y / x ] ph <-> ph )
4	3	imbi1i 237	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
5	1, 4	bitri 183	1 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 104   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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbco2d  1954  sbco2vd  1955  hbsbd  1970