Theorem df3nandALT2 37168

Description: The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion

Ref	Expression
df3nandALT2	⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))

Proof of Theorem df3nandALT2

Step	Hyp	Ref	Expression
1		df-3nand 37166	. 2 ⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))
2		imnan 405	. . 3 ⊢ ((𝜓 → ¬ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒))
3	2	imbi2i 339	. 2 ⊢ ((𝜑 → (𝜓 → ¬ 𝜒)) ↔ (𝜑 → ¬ (𝜓 ∧ 𝜒)))
4		imnan 405	. . 3 ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ¬ (𝜑 ∧ (𝜓 ∧ 𝜒)))
5		3anass 1111	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
6	4, 5	xchbinxr 338	. 2 ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))
7	1, 3, 6	3bitri 300	1 ⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜓 ∧ 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ⊼ w3nand 37165

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-3an 1105  df-3nand 37166

This theorem is used by: (None)