This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem biadan

Description: An implication is equivalent to the equivalence of some implied equivalence and some other equivalence involving a conjunction. A utility lemma as illustrated in biadanii and elelb . (Contributed by BJ, 4-Mar-2023) (Proof shortened by Wolf Lammen, 8-Mar-2023)

Ref Expression
Assertion biadan 
|- ( ( ph -> ps ) <-> ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) )

Proof

Step Hyp Ref Expression
1 pm4.71r 
 |-  ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) )
2 bicom 
 |-  ( ( ph <-> ( ps /\ ph ) ) <-> ( ( ps /\ ph ) <-> ph ) )
3 bicom 
 |-  ( ( ph <-> ( ps /\ ch ) ) <-> ( ( ps /\ ch ) <-> ph ) )
4 pm5.32 
 |-  ( ( ps -> ( ph <-> ch ) ) <-> ( ( ps /\ ph ) <-> ( ps /\ ch ) ) )
5 3 4 bibi12i 
 |-  ( ( ( ph <-> ( ps /\ ch ) ) <-> ( ps -> ( ph <-> ch ) ) ) <-> ( ( ( ps /\ ch ) <-> ph ) <-> ( ( ps /\ ph ) <-> ( ps /\ ch ) ) ) )
6 bicom 
 |-  ( ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) <-> ( ( ph <-> ( ps /\ ch ) ) <-> ( ps -> ( ph <-> ch ) ) ) )
7 biluk 
 |-  ( ( ( ps /\ ph ) <-> ph ) <-> ( ( ( ps /\ ch ) <-> ph ) <-> ( ( ps /\ ph ) <-> ( ps /\ ch ) ) ) )
8 5 6 7 3bitr4ri 
 |-  ( ( ( ps /\ ph ) <-> ph ) <-> ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) )
9 1 2 8 3bitri 
 |-  ( ( ph -> ps ) <-> ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) )