Theorem mpto1 1533

Description: Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mpto2 1534) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)

Hypotheses

Ref	Expression
mpto1.1	|- ph
mpto1.2	|- -. (ph /\ ps)

Assertion

Ref	Expression
mpto1	|- -. ps

Proof of Theorem mpto1

Step	Hyp	Ref	Expression
1		mpto1.1	. 2 |- ph
2		mpto1.2	. . 3 |- -. (ph /\ ps)
3	2	imnani 412	. 2 |- (ph -> -. ps)
4	1, 3	ax-mp 5	1 |- -. ps

This proof depends on syntax axioms:  -. wn 3   /\ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by: (None)