Theorem mptxor 1414

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, 2-Mar-2018.)

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	. . . 4 |- ( ph \/_ ps )
3		df-xor 1366	. . . 4 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
4	2, 3	mpbi 144	. . 3 |- ( ( ph \/ ps ) /\ -. ( ph /\ ps ) )
5	4	simpri 112	. 2 |- -. ( ph /\ ps )
6	1, 5	mptnan 1413	1 |- -. ps

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 698   \/_ wxo 1365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-xor 1366

This theorem is referenced by: (None)