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Theorem ifpan123g 36822
 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 1010 . 2 (if-(𝜑, 𝜒, 𝜏) ↔ ((¬ 𝜑𝜒) ∧ (𝜑𝜏)))
2 dfifp4 1010 . 2 (if-(𝜓, 𝜃, 𝜂) ↔ ((¬ 𝜓𝜃) ∧ (𝜓𝜂)))
31, 2anbi12i 729 1 ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383  if-wif 1006 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 196  df-or 384  df-an 385  df-ifp 1007 This theorem is referenced by:  ifpan23  36823
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