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Theorem rp-fakeinunass 40676
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 40674 . . 3 ((𝑥𝐶𝑥𝐴) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
21albii 1826 . 2 (∀𝑥(𝑥𝐶𝑥𝐴) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
3 dfss2 3863 . 2 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
4 dfcleq 2731 . . 3 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))))
5 elun 4039 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶))
6 elin 3859 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
76orbi1i 913 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
85, 7bitri 278 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
9 elin 3859 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
10 elun 4039 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1110anbi2i 626 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
129, 11bitri 278 . . . . 5 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
138, 12bibi12i 343 . . . 4 ((𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
1413albii 1826 . . 3 (∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
154, 14bitri 278 . 2 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
162, 3, 153bitr4i 306 1 (𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846  wal 1540   = wceq 1542  wcel 2114  cun 3841  cin 3842  wss 3843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-un 3848  df-in 3850  df-ss 3860
This theorem is referenced by:  rp-fakeuninass  40677
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