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Theorem rp-fakeinunass 37342
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 37339 . . 3 ((𝑥𝐶𝑥𝐴) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
21albii 1744 . 2 (∀𝑥(𝑥𝐶𝑥𝐴) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
3 dfss2 3572 . 2 (𝐶𝐴 ↔ ∀𝑥(𝑥𝐶𝑥𝐴))
4 dfcleq 2615 . . 3 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))))
5 elun 3731 . . . . . 6 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶))
6 elin 3774 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
76orbi1i 542 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
85, 7bitri 264 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶))
9 elin 3774 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
10 elun 3731 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1110anbi2i 729 . . . . . 6 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
129, 11bitri 264 . . . . 5 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
138, 12bibi12i 329 . . . 4 ((𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ (((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
1413albii 1744 . . 3 (∀𝑥(𝑥 ∈ ((𝐴𝐵) ∪ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶))) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
154, 14bitri 264 . 2 (((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)) ↔ ∀𝑥(((𝑥𝐴𝑥𝐵) ∨ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶))))
162, 3, 153bitr4i 292 1 (𝐶𝐴 ↔ ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  cun 3553  cin 3554  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-in 3562  df-ss 3569
This theorem is referenced by:  rp-fakeuninass  37343
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