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Theorem naim1 32661
 Description: Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
Assertion
Ref Expression
naim1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Proof of Theorem naim1
StepHypRef Expression
1 con3 149 . . 3 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21orim1d 920 . 2 ((𝜑𝜓) → ((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)))
3 pm3.13 523 . . . 4 (¬ (𝜓𝜒) → (¬ 𝜓 ∨ ¬ 𝜒))
4 pm3.14 524 . . . 4 ((¬ 𝜑 ∨ ¬ 𝜒) → ¬ (𝜑𝜒))
53, 4imim12i 62 . . 3 (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → (¬ (𝜓𝜒) → ¬ (𝜑𝜒)))
6 df-nan 1585 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
7 df-nan 1585 . . 3 ((𝜑𝜒) ↔ ¬ (𝜑𝜒))
85, 6, 73imtr4g 285 . 2 (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → ((𝜓𝜒) → (𝜑𝜒)))
92, 8syl 17 1 ((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ⊼ wnan 1584 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-or 384  df-an 385  df-nan 1585 This theorem is referenced by:  naim1i  32663
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