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Theorem rp-fakeuninass 39388
 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 39387 . 2 (𝐴𝐶 ↔ ((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)))
2 eqcom 2804 . 2 (((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)) ↔ (𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴))
3 incom 4105 . . . 4 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
4 uncom 4056 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54ineq1i 4111 . . . 4 ((𝐵𝐴) ∩ 𝐶) = ((𝐴𝐵) ∩ 𝐶)
63, 5eqtri 2821 . . 3 (𝐶 ∩ (𝐵𝐴)) = ((𝐴𝐵) ∩ 𝐶)
7 uncom 4056 . . . 4 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶𝐵))
8 incom 4105 . . . . 5 (𝐶𝐵) = (𝐵𝐶)
98uneq2i 4063 . . . 4 (𝐴 ∪ (𝐶𝐵)) = (𝐴 ∪ (𝐵𝐶))
107, 9eqtri 2821 . . 3 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵𝐶))
116, 10eqeq12i 2811 . 2 ((𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴) ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
121, 2, 113bitri 298 1 (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   = wceq 1525   ∪ cun 3863   ∩ cin 3864   ⊆ wss 3865 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-un 3870  df-in 3872  df-ss 3880 This theorem is referenced by: (None)
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