Theorem naim2 31555

Description: Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)

Assertion

Ref	Expression
naim2	⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑)))

Proof of Theorem naim2

Step	Hyp	Ref	Expression
1		con3 148	. . 3 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	orim2d 881	. 2 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)))
3		pm3.13 521	. . . 4 ⊢ (¬ (𝜒 ∧ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜓))
4		pm3.14 522	. . . 4 ⊢ ((¬ 𝜒 ∨ ¬ 𝜑) → ¬ (𝜒 ∧ 𝜑))
5	3, 4	imim12i 60	. . 3 ⊢ (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → (¬ (𝜒 ∧ 𝜓) → ¬ (𝜒 ∧ 𝜑)))
6		df-nan 1440	. . 3 ⊢ ((𝜒 ⊼ 𝜓) ↔ ¬ (𝜒 ∧ 𝜓))
7		df-nan 1440	. . 3 ⊢ ((𝜒 ⊼ 𝜑) ↔ ¬ (𝜒 ∧ 𝜑))
8	5, 6, 7	3imtr4g 284	. 2 ⊢ (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑)))
9	2, 8	syl 17	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ⊼ wnan 1439

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 196  df-or 384  df-an 385  df-nan 1440

This theorem is referenced by:  naim2i  31557