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Theorem ifpbi123 38355
 Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
Assertion
Ref Expression
ifpbi123 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))

Proof of Theorem ifpbi123
StepHypRef Expression
1 simp1 1131 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜑𝜓))
2 simp2 1132 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜒𝜃))
31, 2imbi12d 333 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
41notbid 307 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (¬ 𝜑 ↔ ¬ 𝜓))
5 simp3 1133 . . . 4 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜏𝜂))
64, 5imbi12d 333 . . 3 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((¬ 𝜑𝜏) ↔ (¬ 𝜓𝜂)))
73, 6anbi12d 749 . 2 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (((𝜑𝜒) ∧ (¬ 𝜑𝜏)) ↔ ((𝜓𝜃) ∧ (¬ 𝜓𝜂))))
8 dfifp2 1052 . 2 (if-(𝜑, 𝜒, 𝜏) ↔ ((𝜑𝜒) ∧ (¬ 𝜑𝜏)))
9 dfifp2 1052 . 2 (if-(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ∧ (¬ 𝜓𝜂)))
107, 8, 93bitr4g 303 1 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  if-wif 1050   ∧ w3a 1072 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  df-3an 1074 This theorem is referenced by: (None)
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