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	⊢ 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 1366	. . . 4 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
4	2, 3	mpbi 144	. . 3 ⊢ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))
5	4	simpri 112	. 2 ⊢ ¬ (𝜑 ∧ 𝜓)
6	1, 5	mptnan 1413	1 ⊢ ¬ 𝜓

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)