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Theorem cases 1038
 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 892 . . 3 (𝜑 ∨ ¬ 𝜑)
21biantrur 534 . 2 (𝜓 ↔ ((𝜑 ∨ ¬ 𝜑) ∧ 𝜓))
3 andir 1006 . 2 (((𝜑 ∨ ¬ 𝜑) ∧ 𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜓)))
4 cases.1 . . . 4 (𝜑 → (𝜓𝜒))
54pm5.32i 578 . . 3 ((𝜑𝜓) ↔ (𝜑𝜒))
6 cases.2 . . . 4 𝜑 → (𝜓𝜃))
76pm5.32i 578 . . 3 ((¬ 𝜑𝜓) ↔ (¬ 𝜑𝜃))
85, 7orbi12i 912 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
92, 3, 83bitri 300 1 (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by:  casesifp  1074  elimif  4464  elim2if  30355  wl-ifpimpr  35034
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