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Theorem ifpbi13 37653
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi13 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))

Proof of Theorem ifpbi13
StepHypRef Expression
1 simpl 473 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
21imbi1d 331 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜏) ↔ (𝜓𝜏)))
3 notbi 309 . . . . 5 ((𝜑𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
4 imbi12 336 . . . . 5 ((¬ 𝜑 ↔ ¬ 𝜓) → ((𝜒𝜃) → ((¬ 𝜑𝜒) ↔ (¬ 𝜓𝜃))))
53, 4sylbi 207 . . . 4 ((𝜑𝜓) → ((𝜒𝜃) → ((¬ 𝜑𝜒) ↔ (¬ 𝜓𝜃))))
65imp 445 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((¬ 𝜑𝜒) ↔ (¬ 𝜓𝜃)))
72, 6anbi12d 746 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (((𝜑𝜏) ∧ (¬ 𝜑𝜒)) ↔ ((𝜓𝜏) ∧ (¬ 𝜓𝜃))))
8 dfifp2 1013 . 2 (if-(𝜑, 𝜏, 𝜒) ↔ ((𝜑𝜏) ∧ (¬ 𝜑𝜒)))
9 dfifp2 1013 . 2 (if-(𝜓, 𝜏, 𝜃) ↔ ((𝜓𝜏) ∧ (¬ 𝜓𝜃)))
107, 8, 93bitr4g 303 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  if-wif 1011
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
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator