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Theorem rp-fakeinunass 41122
Description: A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.)
Assertion
Ref Expression
rp-fakeinunass (𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))

Proof of Theorem rp-fakeinunass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rp-fakeanorass 41120 . . 3 ((𝑥𝐶𝑥𝐴) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
21albii 1822 . 2 (∀𝑥(𝑥𝐶𝑥𝐴) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
3 dfss2 3907 . 2 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
4 dfcleq 2731 . . 3 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))))
5 elun 4083 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶))
6 elin 3903 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
76orbi1i 911 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
85, 7bitri 274 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
9 elin 3903 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
10 elun 4083 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1110anbi2i 623 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
129, 11bitri 274 . . . . 5 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
138, 12bibi12i 340 . . . 4 ((𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
1413albii 1822 . . 3 (∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
154, 14bitri 274 . 2 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
162, 3, 153bitr4i 303 1 (𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  wal 1537   = wceq 1539  wcel 2106  cun 3885  cin 3886  wss 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904
This theorem is referenced by:  rp-fakeuninass  41123
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