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Theorem rp-fakeuninass 40267
 Description: A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeuninass (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))

Proof of Theorem rp-fakeuninass
StepHypRef Expression
1 rp-fakeinunass 40266 . 2 (𝐴𝐶 ↔ ((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)))
2 eqcom 2805 . 2 (((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)) ↔ (𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴))
3 incom 4128 . . . 4 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
4 uncom 4080 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54ineq1i 4135 . . . 4 ((𝐵𝐴) ∩ 𝐶) = ((𝐴𝐵) ∩ 𝐶)
63, 5eqtri 2821 . . 3 (𝐶 ∩ (𝐵𝐴)) = ((𝐴𝐵) ∩ 𝐶)
7 uncom 4080 . . . 4 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶𝐵))
8 incom 4128 . . . . 5 (𝐶𝐵) = (𝐵𝐶)
98uneq2i 4087 . . . 4 (𝐴 ∪ (𝐶𝐵)) = (𝐴 ∪ (𝐵𝐶))
107, 9eqtri 2821 . . 3 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵𝐶))
116, 10eqeq12i 2813 . 2 ((𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴) ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
121, 2, 113bitri 300 1 (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898 This theorem is referenced by: (None)
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