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Theorem rp-fakeuninass 38469
Description: A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeuninass (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))

Proof of Theorem rp-fakeuninass
StepHypRef Expression
1 rp-fakeinunass 38468 . 2 (𝐴𝐶 ↔ ((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)))
2 eqcom 2771 . 2 (((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)) ↔ (𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴))
3 incom 3966 . . . 4 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
4 uncom 3918 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54ineq1i 3971 . . . 4 ((𝐵𝐴) ∩ 𝐶) = ((𝐴𝐵) ∩ 𝐶)
63, 5eqtri 2786 . . 3 (𝐶 ∩ (𝐵𝐴)) = ((𝐴𝐵) ∩ 𝐶)
7 uncom 3918 . . . 4 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶𝐵))
8 incom 3966 . . . . 5 (𝐶𝐵) = (𝐵𝐶)
98uneq2i 3925 . . . 4 (𝐴 ∪ (𝐶𝐵)) = (𝐴 ∪ (𝐵𝐶))
107, 9eqtri 2786 . . 3 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵𝐶))
116, 10eqeq12i 2778 . 2 ((𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴) ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
121, 2, 113bitri 288 1 (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1652  cun 3729  cin 3730  wss 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-v 3351  df-un 3736  df-in 3738  df-ss 3745
This theorem is referenced by: (None)
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