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Theorem dfxor4 37077
 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 ((𝜑𝜓) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))

Proof of Theorem dfxor4
StepHypRef Expression
1 xor2 1462 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 df-or 384 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
3 imnan 437 . . . 4 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
43bicomi 213 . . 3 (¬ (𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))
52, 4anbi12i 729 . 2 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)))
6 df-an 385 . 2 (((¬ 𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))
71, 5, 63bitri 285 1 ((𝜑𝜓) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ⊻ wxo 1456 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 196  df-or 384  df-an 385  df-xor 1457 This theorem is referenced by:  dfxor5  37078
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