Theorem xornan2 1550

Description: XOR implies NAND (written with the ⊼ connector). (Contributed by BJ, 19-Apr-2019.)

Assertion

Ref	Expression
xornan2	⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓))

Proof of Theorem xornan2

Step	Hyp	Ref	Expression
1		xornan 1549	. 2 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))
2		df-nan 1522	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	sylibr 237	1 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ⊼ wnan 1521   ⊻ wxo 1541

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  df-xor 1542

This theorem is used by: (None)