Theorem 4cases 565

Description: Inference eliminating two antecedents from the four possible cases that result from their true/false combinations.

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	pm2.61an1 364	. 2 |- (ps -> ch)
4		4cases.2	. . 3 |- ((ph /\ -. ps) -> ch)
5		4cases.4	. . 3 |- ((-. ph /\ -. ps) -> ch)
6	4, 5	pm2.61an1 364	. 2 |- (-. ps -> ch)
7	3, 6	pm2.61i 110	1 |- ch

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196

This theorem is referenced by:  suc11reg 3456  znnen 4930

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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