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

Theorem cases 1034
Description: Case disjunction according to the value of 𝜑. (Contributed by NM, 25-Apr-2019.)
Hypotheses
Ref Expression
cases.1 (𝜑 → (𝜓𝜒))
cases.2 𝜑 → (𝜓𝜃))
Assertion
Ref Expression
cases (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))

Proof of Theorem cases
StepHypRef Expression
1 exmid 888 . . 3 (𝜑 ∨ ¬ 𝜑)
21biantrur 531 . 2 (𝜓 ↔ ((𝜑 ∨ ¬ 𝜑) ∧ 𝜓))
3 andir 1002 . 2 (((𝜑 ∨ ¬ 𝜑) ∧ 𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜓)))
4 cases.1 . . . 4 (𝜑 → (𝜓𝜒))
54pm5.32i 575 . . 3 ((𝜑𝜓) ↔ (𝜑𝜒))
6 cases.2 . . . 4 𝜑 → (𝜓𝜃))
76pm5.32i 575 . . 3 ((¬ 𝜑𝜓) ↔ (¬ 𝜑𝜃))
85, 7orbi12i 908 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
92, 3, 83bitri 298 1 (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842
This theorem is referenced by:  casesifp  1068  elimif  4499  elim2if  30226
  Copyright terms: Public domain W3C validator