Theorem abnotataxb 44037

Description: Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Hypotheses

Ref	Expression
abnotataxb.1	⊢ ¬ 𝜑
abnotataxb.2	⊢ 𝜓

Assertion

Ref	Expression
abnotataxb	⊢ (𝜑 ⊻ 𝜓)

Proof of Theorem abnotataxb

Step	Hyp	Ref	Expression
1		abnotataxb.2	. . . . 5 ⊢ 𝜓
2		abnotataxb.1	. . . . 5 ⊢ ¬ 𝜑
3	1, 2	pm3.2i 474	. . . 4 ⊢ (𝜓 ∧ ¬ 𝜑)
4	3	olci 866	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))
5		xor 1015	. . 3 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
6	4, 5	mpbir 234	. 2 ⊢ ¬ (𝜑 ↔ 𝜓)
7		df-xor 1508	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
8	6, 7	mpbir 234	1 ⊢ (𝜑 ⊻ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 847   ⊻ wxo 1507

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 210  df-an 400  df-or 848  df-xor 1508

This theorem is referenced by:  aisfbistiaxb  44041