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

Theorem nabctnabc 41419
 Description: not ( a -> ( b /\ c ) ) we can show: not a implies ( b /\ c ). (Contributed by Jarvin Udandy, 7-Sep-2020.)
Hypothesis
Ref Expression
nabctnabc.1 ¬ (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
nabctnabc 𝜑 → (𝜓𝜒))

Proof of Theorem nabctnabc
StepHypRef Expression
1 nabctnabc.1 . . . . . . . 8 ¬ (𝜑 → (𝜓𝜒))
2 pm4.61 441 . . . . . . . . 9 (¬ (𝜑 → (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒)))
32biimpi 206 . . . . . . . 8 (¬ (𝜑 → (𝜓𝜒)) → (𝜑 ∧ ¬ (𝜓𝜒)))
41, 3ax-mp 5 . . . . . . 7 (𝜑 ∧ ¬ (𝜓𝜒))
54simpli 473 . . . . . 6 𝜑
64simpri 477 . . . . . 6 ¬ (𝜓𝜒)
75, 62th 254 . . . . 5 (𝜑 ↔ ¬ (𝜓𝜒))
8 bicom 212 . . . . . 6 ((𝜑 ↔ ¬ (𝜓𝜒)) ↔ (¬ (𝜓𝜒) ↔ 𝜑))
98biimpi 206 . . . . 5 ((𝜑 ↔ ¬ (𝜓𝜒)) → (¬ (𝜓𝜒) ↔ 𝜑))
107, 9ax-mp 5 . . . 4 (¬ (𝜓𝜒) ↔ 𝜑)
1110biimpi 206 . . 3 (¬ (𝜓𝜒) → 𝜑)
1211con3i 150 . 2 𝜑 → ¬ ¬ (𝜓𝜒))
1312notnotrd 128 1 𝜑 → (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383 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 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator