Theorem sbrim 2007

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 2004	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbrim.1	. . . 4 |- ( ph -> A. x ph )
3	2	sbh 1822	. . 3 |- ( [ y / x ] ph <-> ph )
4	3	imbi1i 238	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
5	1, 4	bitri 184	1 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   [wsb 1808

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbco2d  2017  sbco2vd  2018  hbsbd  2033