Theorem dfxor5 40105

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

Assertion

Ref	Expression
dfxor5	⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓)))

Proof of Theorem dfxor5

Step	Hyp	Ref	Expression
1		dfxor4 40104	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)))
2		con2b 362	. 2 ⊢ (((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)) ↔ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓)))
3	1, 2	xchbinx 336	1 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ⊻ wxo 1500

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 209  df-an 399  df-or 844  df-xor 1501

This theorem is referenced by: (None)