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Theorem ifpdfbi 37637
Description: Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
Assertion
Ref Expression
ifpdfbi ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Proof of Theorem ifpdfbi
StepHypRef Expression
1 dfbi2 659 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
2 ifpim1 37632 . . . . 5 ((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
3 ifpn 1021 . . . . 5 (if-(𝜑, 𝜓, ⊤) ↔ if-(¬ 𝜑, ⊤, 𝜓))
42, 3bitr4i 267 . . . 4 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊤))
5 ifpim2 37635 . . . 4 ((𝜓𝜑) ↔ if-(𝜑, ⊤, ¬ 𝜓))
64, 5anbi12i 732 . . 3 (((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (if-(𝜑, 𝜓, ⊤) ∧ if-(𝜑, ⊤, ¬ 𝜓)))
7 ifpan23 37623 . . . 4 ((if-(𝜑, 𝜓, ⊤) ∧ if-(𝜑, ⊤, ¬ 𝜓)) ↔ if-(𝜑, (𝜓 ∧ ⊤), (⊤ ∧ ¬ 𝜓)))
8 ancom 466 . . . . . 6 ((𝜓 ∧ ⊤) ↔ (⊤ ∧ 𝜓))
9 truan 1499 . . . . . 6 ((⊤ ∧ 𝜓) ↔ 𝜓)
108, 9bitri 264 . . . . 5 ((𝜓 ∧ ⊤) ↔ 𝜓)
11 truan 1499 . . . . 5 ((⊤ ∧ ¬ 𝜓) ↔ ¬ 𝜓)
12 ifpbi23 37636 . . . . 5 ((((𝜓 ∧ ⊤) ↔ 𝜓) ∧ ((⊤ ∧ ¬ 𝜓) ↔ ¬ 𝜓)) → (if-(𝜑, (𝜓 ∧ ⊤), (⊤ ∧ ¬ 𝜓)) ↔ if-(𝜑, 𝜓, ¬ 𝜓)))
1310, 11, 12mp2an 707 . . . 4 (if-(𝜑, (𝜓 ∧ ⊤), (⊤ ∧ ¬ 𝜓)) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
147, 13bitri 264 . . 3 ((if-(𝜑, 𝜓, ⊤) ∧ if-(𝜑, ⊤, ¬ 𝜓)) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
156, 14bitri 264 . 2 (((𝜑𝜓) ∧ (𝜓𝜑)) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
161, 15bitri 264 1 ((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  if-wif 1011  wtru 1482
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-or 385  df-an 386  df-ifp 1012  df-tru 1484
This theorem is referenced by:  ifpbiidcor  37638  ifpbicor  37639
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