Theorem 4casesdan 916

Description: Deduction eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 19-Mar-2013.)

Hypotheses

Ref	Expression
4casesdan.1	|- ((ph /\ (ps /\ ch)) -> th)
4casesdan.2	|- ((ph /\ (ps /\ -. ch)) -> th)
4casesdan.3	|- ((ph /\ (-. ps /\ ch)) -> th)
4casesdan.4	|- ((ph /\ (-. ps /\ -. ch)) -> th)

Assertion

Ref	Expression
4casesdan	|- (ph -> th)

Proof of Theorem 4casesdan

Step	Hyp	Ref	Expression
1		4casesdan.1	. . 3 |- ((ph /\ (ps /\ ch)) -> th)
2	1	expcom 424	. 2 |- ((ps /\ ch) -> (ph -> th))
3		4casesdan.2	. . 3 |- ((ph /\ (ps /\ -. ch)) -> th)
4	3	expcom 424	. 2 |- ((ps /\ -. ch) -> (ph -> th))
5		4casesdan.3	. . 3 |- ((ph /\ (-. ps /\ ch)) -> th)
6	5	expcom 424	. 2 |- ((-. ps /\ ch) -> (ph -> th))
7		4casesdan.4	. . 3 |- ((ph /\ (-. ps /\ -. ch)) -> th)
8	7	expcom 424	. 2 |- ((-. ps /\ -. ch) -> (ph -> th))
9	2, 4, 6, 8	4cases 915	1 |- (ph -> th)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358

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 177  df-an 360

This theorem is referenced by: (None)