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Theorem bisym1 33785
Description: A symmetry with .

See negsym1 33783 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

Assertion
Ref Expression
bisym1 ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓𝜑))

Proof of Theorem bisym1
StepHypRef Expression
1 nbfal 1553 . . 3 𝜓 ↔ (𝜓 ↔ ⊥))
21bibi2i 341 . 2 ((𝜓 ↔ ¬ 𝜓) ↔ (𝜓 ↔ (𝜓 ↔ ⊥)))
3 pm5.19 391 . . 3 ¬ (𝜓 ↔ ¬ 𝜓)
43pm2.21i 119 . 2 ((𝜓 ↔ ¬ 𝜓) → (𝜓𝜑))
52, 4sylbir 238 1 ((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wfal 1550
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-tru 1541  df-fal 1551
This theorem is referenced by: (None)
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