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Theorem dfxor4 36974
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 1461 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 df-or 383 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
3 imnan 436 . . . 4 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
43bicomi 212 . . 3 (¬ (𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))
52, 4anbi12i 728 . 2 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)))
6 df-an 384 . 2 (((¬ 𝜑𝜓) ∧ (𝜑 → ¬ 𝜓)) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))
71, 5, 63bitri 284 1 ((𝜑𝜓) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  wxo 1455
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 195  df-or 383  df-an 384  df-xor 1456
This theorem is referenced by:  dfxor5  36975
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