Theorem xornan

Description: Exclusive disjunction implies alternative denial ("XOR implies NAND"). (Contributed by BJ, 19-Apr-2019)

	Ref	Expression
Assertion	xornan	⊢ ( ( 𝜑 ⊻ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		xor2	⊢ ( ( 𝜑 ⊻ 𝜓 ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ ( 𝜑 ∧ 𝜓 ) ) )
2	1	simprbi	⊢ ( ( 𝜑 ⊻ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜓 ) )