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Theorem abnotataxb 43506
 Description: Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
Hypotheses
Ref Expression
abnotataxb.1 ¬ 𝜑
abnotataxb.2 𝜓
Assertion
Ref Expression
abnotataxb (𝜑𝜓)

Proof of Theorem abnotataxb
StepHypRef Expression
1 abnotataxb.2 . . . . 5 𝜓
2 abnotataxb.1 . . . . 5 ¬ 𝜑
31, 2pm3.2i 474 . . . 4 (𝜓 ∧ ¬ 𝜑)
43olci 863 . . 3 ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))
5 xor 1012 . . 3 (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
64, 5mpbir 234 . 2 ¬ (𝜑𝜓)
7 df-xor 1503 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
86, 7mpbir 234 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   ⊻ wxo 1502 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 210  df-an 400  df-or 845  df-xor 1503 This theorem is referenced by:  aisfbistiaxb  43510
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