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Theorem abnotataxb 41600
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 456 . . . 4 (𝜓 ∧ ¬ 𝜑)
43olci 855 . . 3 ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))
5 xor 1000 . . 3 (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
64, 5mpbir 221 . 2 ¬ (𝜑𝜓)
7 df-xor 1613 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
86, 7mpbir 221 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 836  wxo 1612
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 197  df-an 383  df-or 837  df-xor 1613
This theorem is referenced by:  aisfbistiaxb  41604
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