Theorem 4cases 915

Description: Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)

Hypotheses

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

Assertion

Ref	Expression
4cases	|- ch

Proof of Theorem 4cases

Step	Hyp	Ref	Expression
1		4cases.1	. . 3 |- ((ph /\ ps) -> ch)
2		4cases.3	. . 3 |- ((-. ph /\ ps) -> ch)
3	1, 2	pm2.61ian 765	. 2 |- (ps -> ch)
4		4cases.2	. . 3 |- ((ph /\ -. ps) -> ch)
5		4cases.4	. . 3 |- ((-. ph /\ -. ps) -> ch)
6	4, 5	pm2.61ian 765	. 2 |- (-. ps -> ch)
7	3, 6	pm2.61i 156	1 |- ch

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:  4casesdan  916  ax11eq  2193  ax11el  2194