Theorem xorbin 1315

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 1307	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2		imnan 656	. . . . 5 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	2	biimpri 131	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜑 → ¬ 𝜓))
4	3	adantl 271	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) → (𝜑 → ¬ 𝜓))
5	1, 4	sylbi 119	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 → ¬ 𝜓))
6		pm2.53 673	. . . . 5 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑))
7	6	orcoms 681	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑))
8	7	adantr 270	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) → (¬ 𝜓 → 𝜑))
9	1, 8	sylbi 119	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (¬ 𝜓 → 𝜑))
10	5, 9	impbid 127	1 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ⊻ wxo 1306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-xor 1307

This theorem is referenced by:  xornbi  1317  zeo4  10269  odd2np1  10272