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Theorem rp-fakeuninass 36705
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 36704 . 2 (𝐴𝐶 ↔ ((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)))
2 eqcom 2616 . 2 (((𝐶𝐵) ∪ 𝐴) = (𝐶 ∩ (𝐵𝐴)) ↔ (𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴))
3 incom 3766 . . . 4 (𝐶 ∩ (𝐵𝐴)) = ((𝐵𝐴) ∩ 𝐶)
4 uncom 3718 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54ineq1i 3771 . . . 4 ((𝐵𝐴) ∩ 𝐶) = ((𝐴𝐵) ∩ 𝐶)
63, 5eqtri 2631 . . 3 (𝐶 ∩ (𝐵𝐴)) = ((𝐴𝐵) ∩ 𝐶)
7 uncom 3718 . . . 4 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐶𝐵))
8 incom 3766 . . . . 5 (𝐶𝐵) = (𝐵𝐶)
98uneq2i 3725 . . . 4 (𝐴 ∪ (𝐶𝐵)) = (𝐴 ∪ (𝐵𝐶))
107, 9eqtri 2631 . . 3 ((𝐶𝐵) ∪ 𝐴) = (𝐴 ∪ (𝐵𝐶))
116, 10eqeq12i 2623 . 2 ((𝐶 ∩ (𝐵𝐴)) = ((𝐶𝐵) ∪ 𝐴) ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
121, 2, 113bitri 284 1 (𝐴𝐶 ↔ ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  cun 3537  cin 3538  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-un 3544  df-in 3546  df-ss 3553
This theorem is referenced by: (None)
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