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Theorem elunant 4113
Description: A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
Assertion
Ref Expression
elunant ((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))

Proof of Theorem elunant
StepHypRef Expression
1 elun 4084 . . 3 (𝐶 ∈ (𝐴𝐵) ↔ (𝐶𝐴𝐶𝐵))
21imbi1i 350 . 2 ((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝐶𝐵) → 𝜑))
3 jaob 959 . 2 (((𝐶𝐴𝐶𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))
42, 3bitri 274 1 ((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  wcel 2106  cun 3886
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 3433  df-un 3893
This theorem is referenced by:  unss  4119  ralunb  4126  intun  4913  srcmpltd  33051
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