Theorem xornan2 1473

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 1472	. 2 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))
2		df-nan 1448	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	sylibr 224	1 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ⊼ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ⊼ wnan 1447   ⊻ wxo 1464

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 197  df-or 385  df-an 386  df-nan 1448  df-xor 1465

This theorem is referenced by: (None)