Theorem sbrim 2396

Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sbrim.1	|- F/ x ph

Assertion

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

Proof of Theorem sbrim

Step	Hyp	Ref	Expression
1		sbim 2395	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbrim.1	. . . 4 |- F/ x ph
3	2	sbf 2380	. . 3 |- ( [ y / x ] ph <-> ph )
4	3	imbi1i 339	. 2 |- ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
5	1, 4	bitri 264	1 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbied  2409  sbco2d  2416