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Theorem rp-fakeanorass 37336
Description: A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.)
Assertion
Ref Expression
rp-fakeanorass ((𝜒𝜑) ↔ (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))

Proof of Theorem rp-fakeanorass
StepHypRef Expression
1 pm1.4 401 . . . . . . . 8 ((𝜑𝜒) → (𝜒𝜑))
21ord 392 . . . . . . 7 ((𝜑𝜒) → (¬ 𝜒𝜑))
3 pm4.83 969 . . . . . . . 8 (((𝜒𝜑) ∧ (¬ 𝜒𝜑)) ↔ 𝜑)
43biimpi 206 . . . . . . 7 (((𝜒𝜑) ∧ (¬ 𝜒𝜑)) → 𝜑)
52, 4sylan2 491 . . . . . 6 (((𝜒𝜑) ∧ (𝜑𝜒)) → 𝜑)
65ex 450 . . . . 5 ((𝜒𝜑) → ((𝜑𝜒) → 𝜑))
76anim1d 587 . . . 4 ((𝜒𝜑) → (((𝜑𝜒) ∧ (𝜓𝜒)) → (𝜑 ∧ (𝜓𝜒))))
8 orc 400 . . . . 5 (𝜑 → (𝜑𝜒))
98anim1i 591 . . . 4 ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜒) ∧ (𝜓𝜒)))
107, 9jctir 560 . . 3 ((𝜒𝜑) → ((((𝜑𝜒) ∧ (𝜓𝜒)) → (𝜑 ∧ (𝜓𝜒))) ∧ ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜒) ∧ (𝜓𝜒)))))
11 olc 399 . . . . . 6 (𝜒 → (𝜑𝜒))
12 olc 399 . . . . . 6 (𝜒 → (𝜓𝜒))
1311, 12jca 554 . . . . 5 (𝜒 → ((𝜑𝜒) ∧ (𝜓𝜒)))
14 simpl 473 . . . . 5 ((𝜑 ∧ (𝜓𝜒)) → 𝜑)
1513, 14imim12i 62 . . . 4 ((((𝜑𝜒) ∧ (𝜓𝜒)) → (𝜑 ∧ (𝜓𝜒))) → (𝜒𝜑))
1615adantr 481 . . 3 (((((𝜑𝜒) ∧ (𝜓𝜒)) → (𝜑 ∧ (𝜓𝜒))) ∧ ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜒) ∧ (𝜓𝜒)))) → (𝜒𝜑))
1710, 16impbii 199 . 2 ((𝜒𝜑) ↔ ((((𝜑𝜒) ∧ (𝜓𝜒)) → (𝜑 ∧ (𝜓𝜒))) ∧ ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜒) ∧ (𝜓𝜒)))))
18 dfbi2 659 . 2 ((((𝜑𝜒) ∧ (𝜓𝜒)) ↔ (𝜑 ∧ (𝜓𝜒))) ↔ ((((𝜑𝜒) ∧ (𝜓𝜒)) → (𝜑 ∧ (𝜓𝜒))) ∧ ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜒) ∧ (𝜓𝜒)))))
19 ordir 908 . . . 4 (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
2019bicomi 214 . . 3 (((𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ 𝜒))
2120bibi1i 328 . 2 ((((𝜑𝜒) ∧ (𝜓𝜒)) ↔ (𝜑 ∧ (𝜓𝜒))) ↔ (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))
2217, 18, 213bitr2i 288 1 ((𝜒𝜑) ↔ (((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒))))
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:  rp-fakeoranass  37337  rp-fakeinunass  37339
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