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Theorem rp-fakeinunass 38536
Description: A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by Richard Penner, 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 38534 . . 3 ((𝑥𝐶𝑥𝐴) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
21albii 1914 . 2 (∀𝑥(𝑥𝐶𝑥𝐴) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
3 dfss2 3749 . 2 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
4 dfcleq 2759 . . 3 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))))
5 elun 3915 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶))
6 elin 3958 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
76orbi1i 937 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
85, 7bitri 266 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
9 elin 3958 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
10 elun 3915 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1110anbi2i 616 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
129, 11bitri 266 . . . . 5 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
138, 12bibi12i 330 . . . 4 ((𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
1413albii 1914 . . 3 (∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
154, 14bitri 266 . 2 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
162, 3, 153bitr4i 294 1 (𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  wal 1650   = wceq 1652  wcel 2155  cun 3730  cin 3731  wss 3732
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 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
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 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-in 3739  df-ss 3746
This theorem is referenced by:  rp-fakeuninass  38537
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