Theorem mptnan 1693

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 mptxor 1694) 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
mptnan.min	|- ph
mptnan.maj	|- -. ( ph /\ ps )

Assertion

Ref	Expression
mptnan	|- -. ps

Proof of Theorem mptnan

Step	Hyp	Ref	Expression
1		mptnan.min	. 2 |- ph
2		mptnan.maj	. . 3 |- -. ( ph /\ ps )
3	2	imnani 439	. 2 |- ( ph -> -. ps )
4	1, 3	ax-mp 5	1 |- -. ps

Syntax hints:   -. wn 3   /\ wa 384

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 197  df-an 386

This theorem is referenced by:  mptxor  1694  alephsucpw2  8934  aleph1re  14974