Theorem xorneg2 1550

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 1541	. 2 ⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2		pm5.18 384	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
3		xnor 1542	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))
4	1, 2, 3	3bitr2i 302	1 ⊢ ((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ⊻ 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-xor 1541

This theorem is used by:  xorneg1  1551  xorneg  1552