Theorem xoranor 1268

Description: One way of defining exclusive or. Equivalent to df-xor 1267. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)

Assertion

Ref	Expression
xoranor	⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))

Proof of Theorem xoranor

Step	Hyp	Ref	Expression
1		df-xor 1267	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2		ax-ia3 101	. . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
3	2	con3d 561	. . . . . 6 ⊢ (𝜑 → (¬ (𝜑 ∧ 𝜓) → ¬ 𝜓))
4		olc 632	. . . . . 6 ⊢ (¬ 𝜓 → (¬ 𝜑 ∨ ¬ 𝜓))
5	3, 4	syl6 29	. . . . 5 ⊢ (𝜑 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))
6		pm3.21 251	. . . . . . 7 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
7	6	con3d 561	. . . . . 6 ⊢ (𝜓 → (¬ (𝜑 ∧ 𝜓) → ¬ 𝜑))
8		orc 633	. . . . . 6 ⊢ (¬ 𝜑 → (¬ 𝜑 ∨ ¬ 𝜓))
9	7, 8	syl6 29	. . . . 5 ⊢ (𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))
10	5, 9	jaoi 636	. . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))
11	10	imdistani 419	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
12	1, 11	sylbi 114	. 2 ⊢ ((𝜑 ⊻ 𝜓) → ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
13		pm3.14 670	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓))
14	13	anim2i 324	. . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)) → ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
15	14, 1	sylibr 137	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)) → (𝜑 ⊻ 𝜓))
16	12, 15	impbii 117	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ⊻ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by:  excxor  1269  xoror  1270