MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cases2 Structured version   Visualization version   GIF version

Theorem cases2 1043
Description: Case disjunction according to the value of 𝜑. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 28-Feb-2022.)
Assertion
Ref Expression
cases2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Proof of Theorem cases2
StepHypRef Expression
1 pm4.83 1022 . 2 (((𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) ∧ (¬ 𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
2 dedlema 1041 . . . 4 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
32pm5.74i 274 . . 3 ((𝜑𝜓) ↔ (𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
4 dedlemb 1042 . . . 4 𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
54pm5.74i 274 . . 3 ((¬ 𝜑𝜒) ↔ (¬ 𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
63, 5anbi12i 629 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) ∧ (¬ 𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))))
7 ancom 464 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
8 ancom 464 . . 3 ((¬ 𝜑𝜒) ↔ (𝜒 ∧ ¬ 𝜑))
97, 8orbi12i 912 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
101, 6, 93bitr4ri 307 1 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
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:  dfbi3  1045  dfifp2  1060  ifval  4480  ifpidg  40129  ifpim123g  40138
  Copyright terms: Public domain W3C validator