Theorem xornan 1549

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

Assertion

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

Proof of Theorem xornan

Step	Hyp	Ref	Expression
1		xor2 1547	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2	1	simprbi 503	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ⊻ 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-xor 1542

This theorem is used by:  xornan2  1550  mptxor  1802