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Theorem wl-orel12 32961
Description: In a conjunctive normal form a pair of nodes like (𝜑𝜓) ∧ (¬ 𝜑𝜒) eliminates the need of a node (𝜓𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
Assertion
Ref Expression
wl-orel12 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → (𝜓𝜒))

Proof of Theorem wl-orel12
StepHypRef Expression
1 pm2.1 433 . 2 𝜑𝜑)
2 orel1 397 . . . 4 𝜑 → ((𝜑𝜓) → 𝜓))
3 orc 400 . . . 4 (𝜓 → (𝜓𝜒))
42, 3syl6com 37 . . 3 ((𝜑𝜓) → (¬ 𝜑 → (𝜓𝜒)))
5 notnot 136 . . . . 5 (𝜑 → ¬ ¬ 𝜑)
6 orel1 397 . . . . 5 (¬ ¬ 𝜑 → ((¬ 𝜑𝜒) → 𝜒))
75, 6syl 17 . . . 4 (𝜑 → ((¬ 𝜑𝜒) → 𝜒))
8 olc 399 . . . 4 (𝜒 → (𝜓𝜒))
97, 8syl6com 37 . . 3 ((¬ 𝜑𝜒) → (𝜑 → (𝜓𝜒)))
104, 9jaao 531 . 2 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → ((¬ 𝜑𝜑) → (𝜓𝜒)))
111, 10mpi 20 1 (((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  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:  wl-cases2-dnf  32962
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