Theorem xorneg2 1517

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 1507	. 2 ⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2		pm5.18 383	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
3		xnor 1508	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
4	1, 2, 3	3bitr2i 299	1 ⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 205   ⊻ 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-xor 1507

This theorem is referenced by:  xorneg1  1518  xorneg  1519