Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elimifd Structured version   Visualization version   GIF version

Theorem elimifd 30225
Description: Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Hypotheses
Ref Expression
elimifd.1 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))
elimifd.2 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))
Assertion
Ref Expression
elimifd (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))

Proof of Theorem elimifd
StepHypRef Expression
1 exmid 888 . . . 4 (𝜓 ∨ ¬ 𝜓)
21biantrur 531 . . 3 (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒))
32a1i 11 . 2 (𝜑 → (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒)))
4 andir 1002 . . 3 (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜒)))
54a1i 11 . 2 (𝜑 → (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜒))))
6 iftrue 4469 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
7 elimifd.1 . . . . 5 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))
86, 7syl5 34 . . . 4 (𝜑 → (𝜓 → (𝜒𝜃)))
98pm5.32d 577 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
10 iffalse 4472 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
11 elimifd.2 . . . . 5 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))
1210, 11syl5 34 . . . 4 (𝜑 → (¬ 𝜓 → (𝜒𝜏)))
1312pm5.32d 577 . . 3 (𝜑 → ((¬ 𝜓𝜒) ↔ (¬ 𝜓𝜏)))
149, 13orbi12d 912 . 2 (𝜑 → (((𝜓𝜒) ∨ (¬ 𝜓𝜒)) ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
153, 5, 143bitrd 306 1 (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = 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-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-if 4464
This theorem is referenced by:  elim2if  30226
  Copyright terms: Public domain W3C validator