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Theorem nanorxor 37423
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 1439 . 2 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 xor2 1461 . . . 4 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
32rbaibr 943 . . 3 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ (𝜑𝜓)))
42bibi2i 325 . . . 4 (((𝜑𝜓) ↔ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
5 pm4.71 659 . . . . 5 (((𝜑𝜓) → ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
6 simpl 471 . . . . . . . 8 ((𝜑𝜓) → 𝜑)
76orcd 405 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
87con3i 148 . . . . . 6 (¬ (𝜑𝜓) → ¬ (𝜑𝜓))
9 id 22 . . . . . 6 (¬ (𝜑𝜓) → ¬ (𝜑𝜓))
108, 9ja 171 . . . . 5 (((𝜑𝜓) → ¬ (𝜑𝜓)) → ¬ (𝜑𝜓))
115, 10sylbir 223 . . . 4 (((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))) → ¬ (𝜑𝜓))
124, 11sylbi 205 . . 3 (((𝜑𝜓) ↔ (𝜑𝜓)) → ¬ (𝜑𝜓))
133, 12impbii 197 . 2 (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
141, 13bitri 262 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  wnan 1438  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-nan 1439  df-xor 1456
This theorem is referenced by:  undisjrab  37424
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