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Theorem bj-andnotim 32904
Description: Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-andnotim (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))

Proof of Theorem bj-andnotim
StepHypRef Expression
1 imor 832 . . 3 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∨ 𝜒))
2 iman 388 . . . . 5 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
32biimpri 218 . . . 4 (¬ (𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
43orim1i 875 . . 3 ((¬ (𝜑 ∧ ¬ 𝜓) ∨ 𝜒) → ((𝜑𝜓) ∨ 𝜒))
51, 4sylbi 207 . 2 (((𝜑 ∧ ¬ 𝜓) → 𝜒) → ((𝜑𝜓) ∨ 𝜒))
6 pm2.24 122 . . . . 5 (𝜓 → (¬ 𝜓𝜒))
76imim2i 16 . . . 4 ((𝜑𝜓) → (𝜑 → (¬ 𝜓𝜒)))
87impd 396 . . 3 ((𝜑𝜓) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
9 ax-1 6 . . 3 (𝜒 → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
108, 9jaoi 837 . 2 (((𝜑𝜓) ∨ 𝜒) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
115, 10impbii 199 1 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826
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 827
This theorem is referenced by: (None)
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