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Theorem ifbieq12d2 4068
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 4067 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4 ifbieq12d2.1 . . 3 (𝜑 → (𝜓𝜒))
54ifbid 4057 . 2 (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
63, 5eqtrd 2643 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  ifcif 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-un 3544  df-if 4036
This theorem is referenced by:  ofccat  13505  itgeq12dv  29549  sgnneg  29763
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