Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifbieq12i Structured version   Visualization version   GIF version

Theorem ifbieq12i 4062
 Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
ifbieq12i.1 (𝜑𝜓)
ifbieq12i.2 𝐴 = 𝐶
ifbieq12i.3 𝐵 = 𝐷
Assertion
Ref Expression
ifbieq12i if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Proof of Theorem ifbieq12i
StepHypRef Expression
1 ifbieq12i.2 . . 3 𝐴 = 𝐶
2 ifeq1 4040 . . 3 (𝐴 = 𝐶 → if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵))
31, 2ax-mp 5 . 2 if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵)
4 ifbieq12i.1 . . 3 (𝜑𝜓)
5 ifbieq12i.3 . . 3 𝐵 = 𝐷
64, 5ifbieq2i 4060 . 2 if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
73, 6eqtri 2632 1 if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = 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:  cbvditg  23424  sgnneg  29929  binomcxplemdvsum  37576
 Copyright terms: Public domain W3C validator