Theorem xorneg1 1311

Description: ⊻ is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
xorneg1	⊢ ((¬ φ ⊻ ψ) ↔ ¬ (φ ⊻ ψ))

Proof of Theorem xorneg1

Step	Hyp	Ref	Expression
1		df-xor 1305	. 2 ⊢ ((¬ φ ⊻ ψ) ↔ ¬ (¬ φ ↔ ψ))
2		nbbn 347	. . 3 ⊢ ((¬ φ ↔ ψ) ↔ ¬ (φ ↔ ψ))
3	2	con2bii 322	. 2 ⊢ ((φ ↔ ψ) ↔ ¬ (¬ φ ↔ ψ))
4		xnor 1306	. 2 ⊢ ((φ ↔ ψ) ↔ ¬ (φ ⊻ ψ))
5	1, 3, 4	3bitr2i 264	1 ⊢ ((¬ φ ⊻ ψ) ↔ ¬ (φ ⊻ ψ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ⊻ wxo 1304

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 177  df-xor 1305

This theorem is referenced by:  xorneg2  1312  xorneg  1313