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

Theorem ifbi 4553
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Proof of Theorem ifbi
StepHypRef Expression
1 dfbi3 1049 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
2 iftrue 4537 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3 iftrue 4537 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
43eqcomd 2741 . . . 4 (𝜓𝐴 = if(𝜓, 𝐴, 𝐵))
52, 4sylan9eq 2795 . . 3 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
6 iffalse 4540 . . . 4 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
7 iffalse 4540 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
87eqcomd 2741 . . . 4 𝜓𝐵 = if(𝜓, 𝐴, 𝐵))
96, 8sylan9eq 2795 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
105, 9jaoi 857 . 2 (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
111, 10sylbi 217 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  ifcif 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-if 4532
This theorem is referenced by:  ifbid  4554  ifbieq2i  4556  prodeq1i  15949  gsummoncoe1  22328  scmatscm  22535  mulmarep1gsum1  22595  madugsum  22665  mp2pm2mplem4  22831  dchrhash  27330  lgsdi  27393  rpvmasum2  27571  ifnebib  32570  itgeq12i  36188  bj-projval  36979  matunitlindflem2  37604  itg2gt0cn  37662  dedths  38944  dfafv2  47082
  Copyright terms: Public domain W3C validator