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Theorem wl-cases2-dnf 32962
Description: A particular instance of orddi 912 and anddi 913 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 992, and is related to consensus 998. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1012 and dfifp4 1015, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
wl-cases2-dnf (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem wl-cases2-dnf
StepHypRef Expression
1 exmid 431 . . . . 5 (𝜑 ∨ ¬ 𝜑)
21biantrur 527 . . . 4 ((𝜑𝜒) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑𝜒)))
3 orcom 402 . . . . 5 ((¬ 𝜑𝜓) ↔ (𝜓 ∨ ¬ 𝜑))
4 orcom 402 . . . . 5 ((𝜒𝜓) ↔ (𝜓𝜒))
53, 4anbi12i 732 . . . 4 (((¬ 𝜑𝜓) ∧ (𝜒𝜓)) ↔ ((𝜓 ∨ ¬ 𝜑) ∧ (𝜓𝜒)))
62, 5anbi12i 732 . . 3 (((𝜑𝜒) ∧ ((¬ 𝜑𝜓) ∧ (𝜒𝜓))) ↔ (((𝜑 ∨ ¬ 𝜑) ∧ (𝜑𝜒)) ∧ ((𝜓 ∨ ¬ 𝜑) ∧ (𝜓𝜒))))
7 anass 680 . . 3 ((((𝜑𝜒) ∧ (¬ 𝜑𝜓)) ∧ (𝜒𝜓)) ↔ ((𝜑𝜒) ∧ ((¬ 𝜑𝜓) ∧ (𝜒𝜓))))
8 orddi 912 . . 3 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ (((𝜑 ∨ ¬ 𝜑) ∧ (𝜑𝜒)) ∧ ((𝜓 ∨ ¬ 𝜑) ∧ (𝜓𝜒))))
96, 7, 83bitr4ri 293 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ (((𝜑𝜒) ∧ (¬ 𝜑𝜓)) ∧ (𝜒𝜓)))
10 wl-orel12 32961 . . 3 (((𝜑𝜒) ∧ (¬ 𝜑𝜓)) → (𝜒𝜓))
1110pm4.71i 663 . 2 (((𝜑𝜒) ∧ (¬ 𝜑𝜓)) ↔ (((𝜑𝜒) ∧ (¬ 𝜑𝜓)) ∧ (𝜒𝜓)))
12 ancom 466 . 2 (((𝜑𝜒) ∧ (¬ 𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
139, 11, 123bitr2i 288 1 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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: (None)
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