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Theorem bj-bibibi 32248
Description: A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-bibibi (𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))

Proof of Theorem bj-bibibi
StepHypRef Expression
1 pm5.501 356 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
2 bianir 1008 . . . 4 ((𝜓 ∧ (𝜑𝜓)) → 𝜑)
32ex 450 . . 3 (𝜓 → ((𝜑𝜓) → 𝜑))
4 bibif 361 . . . . 5 𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
54con2bid 344 . . . 4 𝜓 → (𝜑 ↔ ¬ (𝜑𝜓)))
65biimprd 238 . . 3 𝜓 → (¬ (𝜑𝜓) → 𝜑))
73, 6bija 370 . 2 ((𝜓 ↔ (𝜑𝜓)) → 𝜑)
81, 7impbii 199 1 (𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))
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  df-an 386
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator