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Theorem pm5.18 371
Description: Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or." (Contributed by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)
Assertion
Ref Expression
pm5.18 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.18
StepHypRef Expression
1 pm5.501 356 . . . 4 (𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
21con1bid 345 . . 3 (𝜑 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ 𝜓))
3 pm5.501 356 . . 3 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
42, 3bitr2d 269 . 2 (𝜑 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
5 nbn2 360 . . . 4 𝜑 → (¬ ¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
65con1bid 345 . . 3 𝜑 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓))
7 nbn2 360 . . 3 𝜑 → (¬ 𝜓 ↔ (𝜑𝜓)))
86, 7bitr2d 269 . 2 𝜑 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
94, 8pm2.61i 176 1 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196
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 197
This theorem is referenced by:  xor3  372  pm5.19  375  pm5.16  933  dfbi3OLD  994  xorneg2  1471
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