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Theorem ifbieq12d2 4496
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 4495 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4 ifbieq12d2.1 . . 3 (𝜑 → (𝜓𝜒))
54ifbid 4485 . 2 (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
63, 5eqtrd 2853 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-un 3938  df-if 4464
This theorem is referenced by:  ofccat  14317  itgeq12dv  31483  sgnneg  31697
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