Theorem xornan 1548

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 1546	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2	1	simprbi 502	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ⊻ wxo 1540

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 401  df-or 861  df-xor 1541

This theorem is used by:  xornan2  1549  mptxor  1798