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Theorem nanorxor 40866
Description: 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
Assertion
Ref Expression
nanorxor ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))

Proof of Theorem nanorxor
StepHypRef Expression
1 df-nan 1483 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 xor2 1509 . . . 4 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
32rbaibr 541 . . 3 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ (𝜑𝜓)))
42bibi2i 341 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
5 pm4.71 561 . . . . 5 (((𝜑𝜓) → ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
6 simpl 486 . . . . . . . 8 ((𝜑𝜓) → 𝜑)
76orcd 870 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
87con3i 157 . . . . . 6 (¬ (𝜑𝜓) → ¬ (𝜑𝜓))
9 id 22 . . . . . 6 (¬ (𝜑𝜓) → ¬ (𝜑𝜓))
108, 9ja 189 . . . . 5 (((𝜑𝜓) → ¬ (𝜑𝜓)) → ¬ (𝜑𝜓))
115, 10sylbir 238 . . . 4 (((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))) → ¬ (𝜑𝜓))
124, 11sylbi 220 . . 3 (((𝜑𝜓) ↔ (𝜑𝜓)) → ¬ (𝜑𝜓))
133, 12impbii 212 . 2 (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
141, 13bitri 278 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  wnan 1482  wxo 1502
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 210  df-an 400  df-or 845  df-nan 1483  df-xor 1503
This theorem is referenced by:  undisjrab  40867
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