Theorem xorneg2 1312

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

Assertion

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

Proof of Theorem xorneg2

Step	Hyp	Ref	Expression
1		xorneg1 1311	. 2 ⊢ ((¬ ψ ⊻ φ) ↔ ¬ (ψ ⊻ φ))
2		xorcom 1307	. 2 ⊢ ((φ ⊻ ¬ ψ) ↔ (¬ ψ ⊻ φ))
3		xorcom 1307	. . 3 ⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ))
4	3	notbii 287	. 2 ⊢ (¬ (φ ⊻ ψ) ↔ ¬ (ψ ⊻ φ))
5	1, 2, 4	3bitr4i 268	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:  xorneg  1313  hadnot  1393