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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 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))

Proof of Theorem xoranor
StepHypRef Expression
1 df-xor 1267 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 ax-ia3 101 . . . . . . 7 (𝜑 → (𝜓 → (𝜑𝜓)))
32con3d 561 . . . . . 6 (𝜑 → (¬ (𝜑𝜓) → ¬ 𝜓))
4 olc 632 . . . . . 6 𝜓 → (¬ 𝜑 ∨ ¬ 𝜓))
53, 4syl6 29 . . . . 5 (𝜑 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))
6 pm3.21 251 . . . . . . 7 (𝜓 → (𝜑 → (𝜑𝜓)))
76con3d 561 . . . . . 6 (𝜓 → (¬ (𝜑𝜓) → ¬ 𝜑))
8 orc 633 . . . . . 6 𝜑 → (¬ 𝜑 ∨ ¬ 𝜓))
97, 8syl6 29 . . . . 5 (𝜓 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))
105, 9jaoi 636 . . . 4 ((𝜑𝜓) → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))
1110imdistani 419 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
121, 11sylbi 114 . 2 ((𝜑𝜓) → ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
13 pm3.14 670 . . . 4 ((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))
1413anim2i 324 . . 3 (((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)) → ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
1514, 1sylibr 137 . 2 (((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)) → (𝜑𝜓))
1612, 15impbii 117 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
 Colors of variables: wff set class 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
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