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Theorem ifbieq12d2 4520
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 4519 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4 ifbieq12d2.1 . . 3 (𝜑 → (𝜓𝜒))
54ifbid 4509 . 2 (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
63, 5eqtrd 2776 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  ifcif 4486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-un 3915  df-if 4487
This theorem is referenced by:  ofccat  14853  itgeq12dv  32917  sgnneg  33131
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