Theorem imnand2 34518

Description: An → nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)

Assertion

Ref	Expression
imnand2	⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))

Proof of Theorem imnand2

Step	Hyp	Ref	Expression
1		nannot 1491	. . . 4 ⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑))
2		nannot 1491	. . . 4 ⊢ (¬ 𝜓 ↔ (𝜓 ⊼ 𝜓))
3	1, 2	anbi12i 626	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓)))
4	3	notbii 319	. 2 ⊢ (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓)))
5		iman 401	. 2 ⊢ ((¬ 𝜑 → 𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))
6		df-nan 1484	. 2 ⊢ (((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)) ↔ ¬ ((𝜑 ⊼ 𝜑) ∧ (𝜓 ⊼ 𝜓)))
7	4, 5, 6	3bitr4i 302	1 ⊢ ((¬ 𝜑 → 𝜓) ↔ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ⊼ wnan 1483

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-nan 1484

This theorem is referenced by: (None)