Theorem xornan 1515

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 1513	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2	1	simprbi 497	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   ⊻ wxo 1506

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 206  df-an 397  df-or 845  df-xor 1507

This theorem is referenced by:  xornan2  1516  mptxor  1772