Theorem rp-fakeoranass

Description: A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020)

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

Proof

Step	Hyp	Ref	Expression
1		rp-fakeanorass	|- ( ( ph -> ch ) <-> ( ( ( ch /\ ps ) \/ ph ) <-> ( ch /\ ( ps \/ ph ) ) ) )
2		bicom	|- ( ( ( ( ch /\ ps ) \/ ph ) <-> ( ch /\ ( ps \/ ph ) ) ) <-> ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ch /\ ps ) \/ ph ) ) )
3		orcom	|- ( ( ps \/ ph ) <-> ( ph \/ ps ) )
4	3	anbi1ci	|- ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ph \/ ps ) /\ ch ) )
5		orcom	|- ( ( ( ch /\ ps ) \/ ph ) <-> ( ph \/ ( ch /\ ps ) ) )
6		ancom	|- ( ( ch /\ ps ) <-> ( ps /\ ch ) )
7	6	orbi2i	|- ( ( ph \/ ( ch /\ ps ) ) <-> ( ph \/ ( ps /\ ch ) ) )
8	5 7	bitri	|- ( ( ( ch /\ ps ) \/ ph ) <-> ( ph \/ ( ps /\ ch ) ) )
9	4 8	bibi12i	|- ( ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ch /\ ps ) \/ ph ) ) <-> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph \/ ( ps /\ ch ) ) ) )
10	2 9	bitri	|- ( ( ( ( ch /\ ps ) \/ ph ) <-> ( ch /\ ( ps \/ ph ) ) ) <-> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph \/ ( ps /\ ch ) ) ) )
11	1 10	bitri	|- ( ( ph -> ch ) <-> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph \/ ( ps /\ ch ) ) ) )