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Theorem naim1 34234
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 156 . . 3 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21orim1d 965 . 2 ((𝜑𝜓) → ((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)))
3 pm3.13 994 . . . 4 (¬ (𝜓𝜒) → (¬ 𝜓 ∨ ¬ 𝜒))
4 pm3.14 995 . . . 4 ((¬ 𝜑 ∨ ¬ 𝜒) → ¬ (𝜑𝜒))
53, 4imim12i 62 . . 3 (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → (¬ (𝜓𝜒) → ¬ (𝜑𝜒)))
6 df-nan 1487 . . 3 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
7 df-nan 1487 . . 3 ((𝜑𝜒) ↔ ¬ (𝜑𝜒))
85, 6, 73imtr4g 299 . 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  wa 399  wo 846  wnan 1486
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 210  df-an 400  df-or 847  df-nan 1487
This theorem is referenced by:  naim1i  34236
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