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Theorem elunant 4138
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 4109 . . 3 (𝐶 ∈ (𝐴𝐵) ↔ (𝐶𝐴𝐶𝐵))
21imbi1i 353 . 2 ((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝐶𝐵) → 𝜑))
3 jaob 959 . 2 (((𝐶𝐴𝐶𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))
42, 3bitri 278 1 ((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  wcel 2115  cun 3916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-un 3923
This theorem is referenced by:  unss  4144  ralunb  4151  intun  4891  srcmpltd  32366
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