Theorem dfxor4 38058

Description: Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)

Assertion

Ref	Expression
dfxor4	⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)))

Proof of Theorem dfxor4

Step	Hyp	Ref	Expression
1		xor2 1470	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)))
2		df-or 385	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
3		imnan 438	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
4	3	bicomi 214	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (𝜑 → ¬ 𝜓))
5	2, 4	anbi12i 733	. 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)))
6		df-an 386	. 2 ⊢ (((¬ 𝜑 → 𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)))
7	1, 5, 6	3bitri 286	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ⊻ wxo 1464

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 197  df-or 385  df-an 386  df-xor 1465

This theorem is referenced by:  dfxor5  38059