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Theorem cases2 1034
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 1008 . 2 (((𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) ∧ (¬ 𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))) ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
2 dedlema 1032 . . . 4 (𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
32pm5.74i 260 . . 3 ((𝜑𝜓) ↔ (𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
4 dedlemb 1033 . . . 4 𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
54pm5.74i 260 . . 3 ((¬ 𝜑𝜒) ↔ (¬ 𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
63, 5anbi12i 735 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) ↔ ((𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))) ∧ (¬ 𝜑 → ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))))
7 ancom 465 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
8 ancom 465 . . 3 ((¬ 𝜑𝜒) ↔ (𝜒 ∧ ¬ 𝜑))
97, 8orbi12i 544 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑)))
101, 6, 93bitr4ri 293 1 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383
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 197  df-or 384  df-an 385
This theorem is referenced by:  dfbi3  1036  dfifp2  1052  ifval  4271  ifpidg  38338  ifpim123g  38347
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