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Theorem ifpbi123 39734
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 1128 . 2 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜑𝜓))
2 simp2 1129 . 2 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜒𝜃))
3 simp3 1130 . 2 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (𝜏𝜂))
41, 2, 3ifpbi123d 1069 1 (((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  if-wif 1054  w3a 1079
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 208  df-an 397  df-or 842  df-ifp 1055  df-3an 1081
This theorem is referenced by: (None)
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