Theorem xornan 1613

Description: 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 1611	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2	1	simprbi 483	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ⊻ wxo 1605

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 384  df-an 385  df-xor 1606

This theorem is referenced by:  xornan2  1614  mptxor  1835