Theorem naim1 34606

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

Assertion

Ref	Expression
naim1	⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒)))

Proof of Theorem naim1

Step	Hyp	Ref	Expression
1		con3 153	. . 3 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	orim1d 962	. 2 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)))
3		pm3.13 991	. . . 4 ⊢ (¬ (𝜓 ∧ 𝜒) → (¬ 𝜓 ∨ ¬ 𝜒))
4		pm3.14 992	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜒))
5	3, 4	imim12i 62	. . 3 ⊢ (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → (¬ (𝜓 ∧ 𝜒) → ¬ (𝜑 ∧ 𝜒)))
6		df-nan 1486	. . 3 ⊢ ((𝜓 ⊼ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒))
7		df-nan 1486	. . 3 ⊢ ((𝜑 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜒))
8	5, 6, 7	3imtr4g 295	. 2 ⊢ (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒)))
9	2, 8	syl 17	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ⊼ wnan 1485

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 206  df-an 396  df-or 844  df-nan 1486

This theorem is referenced by:  naim1i  34608