Theorem wl-sblimt 31949

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2245. (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 2244	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbft 2228	. . 3 |- ( F/ x ps -> ( [ y / x ] ps <-> ps ) )
3	2	imbi2d 323	. 2 |- ( F/ x ps -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [ y / x ] ph -> ps ) ) )
4	1, 3	syl5bb 265	1 |- ( F/ x ps -> ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 189   F/wnf 1675   [wsb 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806

This theorem is referenced by: (None)