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Theorem rp-fakeoranass 39873
Description: A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeoranass ((𝜑𝜒) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))

Proof of Theorem rp-fakeoranass
StepHypRef Expression
1 rp-fakeanorass 39872 . 2 ((𝜑𝜒) ↔ (((𝜒𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓𝜑))))
2 bicom 224 . . 3 ((((𝜒𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓𝜑))) ↔ ((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜒𝜓) ∨ 𝜑)))
3 orcom 866 . . . . 5 ((𝜓𝜑) ↔ (𝜑𝜓))
43anbi1ci 627 . . . 4 ((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜑𝜓) ∧ 𝜒))
5 orcom 866 . . . . 5 (((𝜒𝜓) ∨ 𝜑) ↔ (𝜑 ∨ (𝜒𝜓)))
6 ancom 463 . . . . . 6 ((𝜒𝜓) ↔ (𝜓𝜒))
76orbi2i 909 . . . . 5 ((𝜑 ∨ (𝜒𝜓)) ↔ (𝜑 ∨ (𝜓𝜒)))
85, 7bitri 277 . . . 4 (((𝜒𝜓) ∨ 𝜑) ↔ (𝜑 ∨ (𝜓𝜒)))
94, 8bibi12i 342 . . 3 (((𝜒 ∧ (𝜓𝜑)) ↔ ((𝜒𝜓) ∨ 𝜑)) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
102, 9bitri 277 . 2 ((((𝜒𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓𝜑))) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
111, 10bitri 277 1 ((𝜑𝜒) ↔ (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843
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 209  df-an 399  df-or 844
This theorem is referenced by: (None)
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