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Theorem abnotbtaxb 41725
Description: Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)
Hypotheses
Ref Expression
abnotbtaxb.1 𝜑
abnotbtaxb.2 ¬ 𝜓
Assertion
Ref Expression
abnotbtaxb (𝜑𝜓)

Proof of Theorem abnotbtaxb
StepHypRef Expression
1 abnotbtaxb.1 . . 3 𝜑
2 abnotbtaxb.2 . . 3 ¬ 𝜓
3 xor3 373 . . . 4 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4 pm5.1 854 . . . . . 6 ((𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ ¬ 𝜓))
5 ibibr 359 . . . . . 6 (((𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))))
64, 5mpbi 221 . . . . 5 ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)))
71, 2, 6mp2an 683 . . . 4 ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
83, 7bitri 266 . . 3 (¬ (𝜑𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
91, 2, 8mpbir2an 702 . 2 ¬ (𝜑𝜓)
10 df-xor 1634 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
119, 10mpbir 222 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wxo 1633
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 198  df-an 385  df-xor 1634
This theorem is referenced by:  aistbisfiaxb  41729  aifftbifffaibifff  41732
  Copyright terms: Public domain W3C validator