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Theorem ifpan123g 36719
Description: Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
Assertion
Ref Expression
ifpan123g ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))

Proof of Theorem ifpan123g
StepHypRef Expression
1 dfifp4 1009 . 2 (if-(𝜑, 𝜒, 𝜏) ↔ ((¬ 𝜑𝜒) ∧ (𝜑𝜏)))
2 dfifp4 1009 . 2 (if-(𝜓, 𝜃, 𝜂) ↔ ((¬ 𝜓𝜃) ∧ (𝜓𝜂)))
31, 2anbi12i 728 1 ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wo 381  wa 382  if-wif 1005
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 195  df-or 383  df-an 384  df-ifp 1006
This theorem is referenced by:  ifpan23  36720
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