Theorem condan 769

Description: Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.)

Hypotheses

Ref	Expression
condan.1	|- ((ph /\ -. ps) -> ch)
condan.2	|- ((ph /\ -. ps) -> -. ch)

Assertion

Ref	Expression
condan	|- (ph -> ps)

Proof of Theorem condan

Step	Hyp	Ref	Expression
1		condan.1	. . 3 |- ((ph /\ -. ps) -> ch)
2		condan.2	. . 3 |- ((ph /\ -. ps) -> -. ch)
3	1, 2	pm2.65da 559	. 2 |- (ph -> -. -. ps)
4		notnot2 104	. 2 |- (-. -. ps -> ps)
5	3, 4	syl 15	1 |- (ph -> ps)

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)