Theorem naim2 37178

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 154	. . 3 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	orim2d 982	. 2 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)))
3		pm3.13 1010	. . . 4 ⊢ (¬ (𝜒 ∧ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜓))
4		pm3.14 1011	. . . 4 ⊢ ((¬ 𝜒 ∨ ¬ 𝜑) → ¬ (𝜒 ∧ 𝜑))
5	3, 4	imim12i 63	. . 3 ⊢ (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → (¬ (𝜒 ∧ 𝜓) → ¬ (𝜒 ∧ 𝜑)))
6		df-nan 1522	. . 3 ⊢ ((𝜒 ⊼ 𝜓) ↔ ¬ (𝜒 ∧ 𝜓))
7		df-nan 1522	. . 3 ⊢ ((𝜒 ⊼ 𝜑) ↔ ¬ (𝜒 ∧ 𝜑))
8	5, 6, 7	3imtr4g 299	. 2 ⊢ (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑)))
9	2, 8	syl 18	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ⊼ wnan 1521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-nan 1522

This theorem is used by:  naim2i  37180