Theorem abnotbtaxb 43158

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

Hypotheses

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

Assertion

Ref	Expression
abnotbtaxb	⊢ (𝜑 ⊻ 𝜓)

Proof of Theorem abnotbtaxb

Step	Hyp	Ref	Expression
1		abnotbtaxb.1	. . 3 ⊢ 𝜑
2		abnotbtaxb.2	. . 3 ⊢ ¬ 𝜓
3		xor3 386	. . . 4 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4		pm5.1 821	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ ¬ 𝜓))
5		ibibr 371	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))))
6	4, 5	mpbi 232	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)))
7	1, 2, 6	mp2an 690	. . . 4 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
8	3, 7	bitri 277	. . 3 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
9	1, 2, 8	mpbir2an 709	. 2 ⊢ ¬ (𝜑 ↔ 𝜓)
10		df-xor 1502	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
11	9, 10	mpbir 233	1 ⊢ (𝜑 ⊻ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ⊻ wxo 1501

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 209  df-an 399  df-xor 1502

This theorem is referenced by:  aistbisfiaxb  43162  aifftbifffaibifff  43165