Theorem xorneg2 1510

Description: The connector ⊻ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)

Assertion

Ref	Expression
xorneg2	⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

Proof of Theorem xorneg2

Step	Hyp	Ref	Expression
1		df-xor 1501	. 2 ⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2		pm5.18 385	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
3		xnor 1502	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
4	1, 2, 3	3bitr2i 301	1 ⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1500

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 209  df-xor 1501

This theorem is referenced by:  xorneg1  1511  xorneg  1512