Theorem wl-sbrimt 33331

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2396. (Contributed by Wolf Lammen, 26-Jul-2019.)

Assertion

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

Proof of Theorem wl-sbrimt

Step	Hyp	Ref	Expression
1		sbim 2395	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		sbft 2379	. . 3 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
3	2	imbi1d 331	. 2 |- ( F/ x ph -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( ph -> [ y / x ] ps ) ) )
4	1, 3	syl5bb 272	1 |- ( F/ x ph -> ( [ 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: (None)