Theorem wl-sblimt 31879

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2225. (Contributed by Wolf Lammen, 26-Jul-2019.)

Assertion

Ref	Expression
wl-sblimt	|- ( F/ x ps -> ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) )

Proof of Theorem wl-sblimt

Step	Hyp	Ref	Expression
1		sbim 2224	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbft 2208	. . 3 |- ( F/ x ps -> ( [ y / x ] ps <-> ps ) )
3	2	imbi2d 318	. 2 |- ( F/ x ps -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) ) )
4	1, 3	syl5bb 261	1 |- ( F/ x ps -> ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 188   F/wnf 1667   [wsb 1797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798

This theorem is referenced by: (None)