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Theorem ifpbi12 38353
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi12 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏)))

Proof of Theorem ifpbi12
StepHypRef Expression
1 imbi12 335 . . . 4 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))
21imp 444 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
3 simpl 474 . . . . 5 (((𝜑𝜓) ∧ (𝜒𝜃)) → (𝜑𝜓))
43notbid 307 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃)) → (¬ 𝜑 ↔ ¬ 𝜓))
54imbi1d 330 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃)) → ((¬ 𝜑𝜏) ↔ (¬ 𝜓𝜏)))
62, 5anbi12d 749 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) → (((𝜑𝜒) ∧ (¬ 𝜑𝜏)) ↔ ((𝜓𝜃) ∧ (¬ 𝜓𝜏))))
7 dfifp2 1052 . 2 (if-(𝜑, 𝜒, 𝜏) ↔ ((𝜑𝜒) ∧ (¬ 𝜑𝜏)))
8 dfifp2 1052 . 2 (if-(𝜓, 𝜃, 𝜏) ↔ ((𝜓𝜃) ∧ (¬ 𝜓𝜏)))
96, 7, 83bitr4g 303 1 (((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  if-wif 1050
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 384  df-an 385  df-ifp 1051
This theorem is referenced by: (None)
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