Theorem xorbin 1384

Description: A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)

Assertion

Ref	Expression
xorbin	⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))

Proof of Theorem xorbin

Step	Hyp	Ref	Expression
1		df-xor 1376	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2		imnan 690	. . . . 5 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	2	biimpri 133	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜑 → ¬ 𝜓))
4	3	adantl 277	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) → (𝜑 → ¬ 𝜓))
5	1, 4	sylbi 121	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 → ¬ 𝜓))
6		pm2.53 722	. . . . 5 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑))
7	6	orcoms 730	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑))
8	7	adantr 276	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) → (¬ 𝜓 → 𝜑))
9	1, 8	sylbi 121	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (¬ 𝜓 → 𝜑))
10	5, 9	impbid 129	1 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ⊻ wxo 1375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-xor 1376

This theorem is referenced by:  xornbi  1386  zeo4  11840  odd2np1  11843