Theorem mptxor 1315

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	⊢ 𝜑
mptxor.maj	⊢ (𝜑 ⊻ 𝜓)

Assertion

Ref	Expression
mptxor	⊢ ¬ 𝜓

Proof of Theorem mptxor

Step	Hyp	Ref	Expression
1		mptxor.min	. 2 ⊢ 𝜑
2		mptxor.maj	. . . 4 ⊢ (𝜑 ⊻ 𝜓)
3		df-xor 1267	. . . 4 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
4	2, 3	mpbi 133	. . 3 ⊢ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))
5	4	simpri 106	. 2 ⊢ ¬ (𝜑 ∧ 𝜓)
6	1, 5	mptnan 1314	1 ⊢ ¬ 𝜓

Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 629   ⊻ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by: (None)