Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifbieq12d2 Structured version   Visualization version   GIF version

Theorem ifbieq12d2 4503
 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 4502 . 2 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
4 ifbieq12d2.1 . . 3 (𝜑 → (𝜓𝜒))
54ifbid 4492 . 2 (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷))
63, 5eqtrd 2861 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 1530  ifcif 4470 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-un 3945  df-if 4471 This theorem is referenced by:  ofccat  14319  itgeq12dv  31470  sgnneg  31684
 Copyright terms: Public domain W3C validator