Theorem rp-fakeoranass 37859

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

Assertion

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

Proof of Theorem rp-fakeoranass

Step	Hyp	Ref	Expression
1		rp-fakeanorass 37858	. 2 |- ( ( ph -> ch ) <-> ( ( ( ch /\ ps ) \/ ph ) <-> ( ch /\ ( ps \/ ph ) ) ) )
2		bicom 212	. . 3 |- ( ( ( ( ch /\ ps ) \/ ph ) <-> ( ch /\ ( ps \/ ph ) ) ) <-> ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ch /\ ps ) \/ ph ) ) )
3		ancom 466	. . . . 5 |- ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ps \/ ph ) /\ ch ) )
4		orcom 402	. . . . . 6 |- ( ( ps \/ ph ) <-> ( ph \/ ps ) )
5	4	anbi1i 731	. . . . 5 |- ( ( ( ps \/ ph ) /\ ch ) <-> ( ( ph \/ ps ) /\ ch ) )
6	3, 5	bitri 264	. . . 4 |- ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ph \/ ps ) /\ ch ) )
7		orcom 402	. . . . 5 |- ( ( ( ch /\ ps ) \/ ph ) <-> ( ph \/ ( ch /\ ps ) ) )
8		ancom 466	. . . . . 6 |- ( ( ch /\ ps ) <-> ( ps /\ ch ) )
9	8	orbi2i 541	. . . . 5 |- ( ( ph \/ ( ch /\ ps ) ) <-> ( ph \/ ( ps /\ ch ) ) )
10	7, 9	bitri 264	. . . 4 |- ( ( ( ch /\ ps ) \/ ph ) <-> ( ph \/ ( ps /\ ch ) ) )
11	6, 10	bibi12i 329	. . 3 |- ( ( ( ch /\ ( ps \/ ph ) ) <-> ( ( ch /\ ps ) \/ ph ) ) <-> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph \/ ( ps /\ ch ) ) ) )
12	2, 11	bitri 264	. 2 |- ( ( ( ( ch /\ ps ) \/ ph ) <-> ( ch /\ ( ps \/ ph ) ) ) <-> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph \/ ( ps /\ ch ) ) ) )
13	1, 12	bitri 264	1 |- ( ( ph -> ch ) <-> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph \/ ( ps /\ ch ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386

This theorem is referenced by: (None)