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Theorem ifbieq12d2 4069
 Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
Hypotheses
Ref Expression
ifbieq12d2.1 (𝜑 → (𝜓𝜒))
ifbieq12d2.2 ((𝜑𝜓) → 𝐴 = 𝐶)
ifbieq12d2.3 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
Assertion
Ref Expression
ifbieq12d2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Proof of Theorem ifbieq12d2
StepHypRef Expression
1 ifbieq12d2.2 . . 3 ((𝜑𝜓) → 𝐴 = 𝐶)
2 ifbieq12d2.3 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
31, 2ifeq12da 4068 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4 ifbieq12d2.1 . . 3 (𝜑 → (𝜓𝜒))
54ifbid 4058 . 2 (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
63, 5eqtrd 2644 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037 This theorem is referenced by:  ofccat  13556  itgeq12dv  29715  sgnneg  29929
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