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Theorem cases2 992
Description: Case disjunction according to the value of 𝜑. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)
Assertion
Ref Expression
cases2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Proof of Theorem cases2
StepHypRef Expression
1 pm3.4 583 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
2 pm2.24 121 . . . . 5 (𝜑 → (¬ 𝜑𝜒))
32adantr 481 . . . 4 ((𝜑𝜓) → (¬ 𝜑𝜒))
41, 3jca 554 . . 3 ((𝜑𝜓) → ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
5 pm2.21 120 . . . . 5 𝜑 → (𝜑𝜓))
65adantr 481 . . . 4 ((¬ 𝜑𝜒) → (𝜑𝜓))
7 pm3.4 583 . . . 4 ((¬ 𝜑𝜒) → (¬ 𝜑𝜒))
86, 7jca 554 . . 3 ((¬ 𝜑𝜒) → ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
94, 8jaoi 394 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
10 pm2.27 42 . . . . . 6 (𝜑 → ((𝜑𝜓) → 𝜓))
1110imdistani 725 . . . . 5 ((𝜑 ∧ (𝜑𝜓)) → (𝜑𝜓))
1211orcd 407 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
1312adantrr 752 . . 3 ((𝜑 ∧ ((𝜑𝜓) ∧ (¬ 𝜑𝜒))) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
14 pm2.27 42 . . . . . 6 𝜑 → ((¬ 𝜑𝜒) → 𝜒))
1514imdistani 725 . . . . 5 ((¬ 𝜑 ∧ (¬ 𝜑𝜒)) → (¬ 𝜑𝜒))
1615olcd 408 . . . 4 ((¬ 𝜑 ∧ (¬ 𝜑𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
1716adantrl 751 . . 3 ((¬ 𝜑 ∧ ((𝜑𝜓) ∧ (¬ 𝜑𝜒))) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
1813, 17pm2.61ian 830 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
199, 18impbii 199 1 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384
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 385  df-an 386
This theorem is referenced by:  dfbi3  993  dfifp2  1013  ifval  4118  ifpidg  37655  ifpim123g  37664
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