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Theorem naim2 32722
 Description: Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
Assertion
Ref Expression
naim2 ((𝜑𝜓) → ((𝜒𝜓) → (𝜒𝜑)))

Proof of Theorem naim2
StepHypRef Expression
1 con3 150 . . 3 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21orim2d 951 . 2 ((𝜑𝜓) → ((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)))
3 pm3.13 979 . . . 4 (¬ (𝜒𝜓) → (¬ 𝜒 ∨ ¬ 𝜓))
4 pm3.14 980 . . . 4 ((¬ 𝜒 ∨ ¬ 𝜑) → ¬ (𝜒𝜑))
53, 4imim12i 62 . . 3 (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → (¬ (𝜒𝜓) → ¬ (𝜒𝜑)))
6 df-nan 1596 . . 3 ((𝜒𝜓) ↔ ¬ (𝜒𝜓))
7 df-nan 1596 . . 3 ((𝜒𝜑) ↔ ¬ (𝜒𝜑))
85, 6, 73imtr4g 285 . 2 (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → ((𝜒𝜓) → (𝜒𝜑)))
92, 8syl 17 1 ((𝜑𝜓) → ((𝜒𝜓) → (𝜒𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∨ wo 836   ⊼ wnan 1595 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 837  df-nan 1596 This theorem is referenced by:  naim2i  32724
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