Theorem mptxor 1694

Description: Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or \/_. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)

Hypotheses

Ref	Expression
mptxor.min	|- ph
mptxor.maj	|- ( ph \/_ ps )

Assertion

Ref	Expression
mptxor	|- -. ps

Proof of Theorem mptxor

Step	Hyp	Ref	Expression
1		mptxor.min	. 2 |- ph
2		mptxor.maj	. . 3 |- ( ph \/_ ps )
3		xornan 1472	. . 3 |- ( ( ph \/_ ps ) -> -. ( ph /\ ps ) )
4	2, 3	ax-mp 5	. 2 |- -. ( ph /\ ps )
5	1, 4	mptnan 1693	1 |- -. ps

Syntax hints:   -. wn 3   /\ wa 384   \/_ wxo 1464

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-or 385  df-an 386  df-xor 1465

This theorem is referenced by: (None)