Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifbieq2i Structured version   Visualization version   GIF version

Theorem ifbieq2i 4247
 Description: Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
ifbieq2i.1 (𝜑𝜓)
ifbieq2i.2 𝐴 = 𝐵
Assertion
Ref Expression
ifbieq2i if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Proof of Theorem ifbieq2i
StepHypRef Expression
1 ifbieq2i.1 . . 3 (𝜑𝜓)
2 ifbi 4244 . . 3 ((𝜑𝜓) → if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴))
31, 2ax-mp 5 . 2 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐴)
4 ifbieq2i.2 . . 3 𝐴 = 𝐵
5 ifeq2 4228 . . 3 (𝐴 = 𝐵 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
64, 5ax-mp 5 . 2 if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
73, 6eqtri 2792 1 if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1630  ifcif 4223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-un 3726  df-if 4224 This theorem is referenced by:  ifbieq12i  4249  gcdcom  15442  gcdass  15471  lcmcom  15513  lcmass  15534  bj-xpimasn  33267  cdleme31sdnN  36189  cdlemefr44  36227  cdleme48fv  36301  cdlemeg49lebilem  36341  cdleme50eq  36343  hoidmvlelem3  41325  hoidmvlelem4  41326
 Copyright terms: Public domain W3C validator