Theorem cases 992

Description: Case disjunction according to the value of ph. (Contributed by NM, 25-Apr-2019.)

Hypotheses

Ref	Expression
cases.1	|- ( ph -> ( ps <-> ch ) )
cases.2	|- ( -. ph -> ( ps <-> th ) )

Assertion

Ref	Expression
cases	|- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )

Proof of Theorem cases

Step	Hyp	Ref	Expression
1		exmid 431	. . 3 |- ( ph \/ -. ph )
2	1	biantrur 527	. 2 |- ( ps <-> ( ( ph \/ -. ph ) /\ ps ) )
3		andir 912	. 2 |- ( ( ( ph \/ -. ph ) /\ ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ps ) ) )
4		cases.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
5	4	pm5.32i 669	. . 3 |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
6		cases.2	. . . 4 |- ( -. ph -> ( ps <-> th ) )
7	6	pm5.32i 669	. . 3 |- ( ( -. ph /\ ps ) <-> ( -. ph /\ th ) )
8	5, 7	orbi12i 543	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ps ) ) <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )
9	2, 3, 8	3bitri 286	1 |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )

Syntax hints:   -. wn 3   -> 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:  casesifp  1026  elimif  4122  elim2if  29363