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Theorem ifpim23g 40200
Description: Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
Assertion
Ref Expression
ifpim23g (((𝜑𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑𝜓) → 𝜒) ∧ (𝜒 → (𝜑𝜓))))

Proof of Theorem ifpim23g
StepHypRef Expression
1 ifpidg 40196 . 2 (((𝜑𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ ((((𝜒𝜓) → (𝜑𝜓)) ∧ ((𝜒 ∧ (𝜑𝜓)) → 𝜓)) ∧ ((¬ 𝜑 → (𝜒 ∨ (𝜑𝜓))) ∧ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑)))))
2 dfor2 899 . . . . 5 ((𝜑𝜓) ↔ ((𝜑𝜓) → 𝜓))
32imbi2i 339 . . . 4 ((𝜒 → (𝜑𝜓)) ↔ (𝜒 → ((𝜑𝜓) → 𝜓)))
4 impexp 454 . . . 4 (((𝜒 ∧ (𝜑𝜓)) → 𝜓) ↔ (𝜒 → ((𝜑𝜓) → 𝜓)))
5 ax-1 6 . . . . . 6 (𝜓 → (𝜑𝜓))
65adantl 485 . . . . 5 ((𝜒𝜓) → (𝜑𝜓))
76biantrur 534 . . . 4 (((𝜒 ∧ (𝜑𝜓)) → 𝜓) ↔ (((𝜒𝜓) → (𝜑𝜓)) ∧ ((𝜒 ∧ (𝜑𝜓)) → 𝜓)))
83, 4, 73bitr2i 302 . . 3 ((𝜒 → (𝜑𝜓)) ↔ (((𝜒𝜓) → (𝜑𝜓)) ∧ ((𝜒 ∧ (𝜑𝜓)) → 𝜓)))
9 impexp 454 . . . . 5 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
10 imdi 394 . . . . . 6 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
11 imor 850 . . . . . . . 8 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
12 orcom 867 . . . . . . . 8 ((¬ 𝜑𝜒) ↔ (𝜒 ∨ ¬ 𝜑))
1311, 12bitri 278 . . . . . . 7 ((𝜑𝜒) ↔ (𝜒 ∨ ¬ 𝜑))
1413imbi2i 339 . . . . . 6 (((𝜑𝜓) → (𝜑𝜒)) ↔ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑)))
1510, 14bitri 278 . . . . 5 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑)))
169, 15bitri 278 . . . 4 (((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑)))
17 pm2.21 123 . . . . . 6 𝜑 → (𝜑𝜓))
1817olcd 871 . . . . 5 𝜑 → (𝜒 ∨ (𝜑𝜓)))
1918biantrur 534 . . . 4 (((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑)) ↔ ((¬ 𝜑 → (𝜒 ∨ (𝜑𝜓))) ∧ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑))))
2016, 19bitri 278 . . 3 (((𝜑𝜓) → 𝜒) ↔ ((¬ 𝜑 → (𝜒 ∨ (𝜑𝜓))) ∧ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑))))
218, 20anbi12i 629 . 2 (((𝜒 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜒)) ↔ ((((𝜒𝜓) → (𝜑𝜓)) ∧ ((𝜒 ∧ (𝜑𝜓)) → 𝜓)) ∧ ((¬ 𝜑 → (𝜒 ∨ (𝜑𝜓))) ∧ ((𝜑𝜓) → (𝜒 ∨ ¬ 𝜑)))))
22 ancom 464 . 2 (((𝜒 → (𝜑𝜓)) ∧ ((𝜑𝜓) → 𝜒)) ↔ (((𝜑𝜓) → 𝜒) ∧ (𝜒 → (𝜑𝜓))))
231, 21, 223bitr2i 302 1 (((𝜑𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑𝜓) → 𝜒) ∧ (𝜒 → (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  if-wif 1058
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-an 400  df-or 845  df-ifp 1059
This theorem is referenced by:  ifpim3  40201  ifpim4  40203
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