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Theorem rp-fakeuninass 43529
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 43528 . 2 (𝐴𝐶 ↔ ((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)))
2 eqcom 2744 . 2 (((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)) ↔ (𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴))
3 incom 4209 . . . 4 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
4 uncom 4158 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54ineq1i 4216 . . . 4 ((𝐵𝐴) ∩ 𝐶) = ((𝐴𝐵) ∩ 𝐶)
63, 5eqtri 2765 . . 3 (𝐶 ∩ (𝐵𝐴)) = ((𝐴𝐵) ∩ 𝐶)
7 uncom 4158 . . . 4 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶𝐵))
8 incom 4209 . . . . 5 (𝐶𝐵) = (𝐵𝐶)
98uneq2i 4165 . . . 4 (𝐴 ∪ (𝐶𝐵)) = (𝐴 ∪ (𝐵𝐶))
107, 9eqtri 2765 . . 3 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵𝐶))
116, 10eqeq12i 2755 . 2 ((𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴) ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
121, 2, 113bitri 297 1 (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  cun 3949  cin 3950  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-ss 3968
This theorem is referenced by: (None)
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