Mathbox for Jarvin Udandy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abnotbtaxb Structured version   Visualization version   GIF version

Theorem abnotbtaxb 41607
 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 371 . . . 4 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
4 pm5.1 938 . . . . . 6 ((𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ ¬ 𝜓))
5 ibibr 357 . . . . . 6 (((𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))))
64, 5mpbi 220 . . . . 5 ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓)))
71, 2, 6mp2an 710 . . . 4 ((𝜑 ↔ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
83, 7bitri 264 . . 3 (¬ (𝜑𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
91, 2, 8mpbir2an 993 . 2 ¬ (𝜑𝜓)
10 df-xor 1614 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
119, 10mpbir 221 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ⊻ wxo 1613 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 385  df-xor 1614 This theorem is referenced by:  aistbisfiaxb  41611  aifftbifffaibifff  41614
 Copyright terms: Public domain W3C validator