Theorem xorneg1 1551

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
xorneg1	⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

Proof of Theorem xorneg1

Step	Hyp	Ref	Expression
1		xorcom 1543	. 2 ⊢ ((¬ 𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ ¬ 𝜑))
2		xorneg2 1550	. . 3 ⊢ ((𝜓 ⊻ ¬ 𝜑) ↔ ¬ (𝜓 ⊻ 𝜑))
3		xorcom 1543	. . 3 ⊢ ((𝜓 ⊻ 𝜑) ↔ (𝜑 ⊻ 𝜓))
4	2, 3	xchbinx 337	. 2 ⊢ ((𝜓 ⊻ ¬ 𝜑) ↔ ¬ (𝜑 ⊻ 𝜓))
5	1, 4	bitri 278	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:  xorneg  1552