Theorem sbrim 1985

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 1982	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbrim.1	. . . 4 |- ( ph -> A. x ph )
3	2	sbh 1800	. . 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 1371   [wsb 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbco2d  1995  sbco2vd  1996  hbsbd  2011