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Theorem iunssd 4941
 Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
iunssd.1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
iunssd (𝜑 𝑥𝐴 𝐵𝐶)
Distinct variable groups:   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iunssd
StepHypRef Expression
1 iunssd.1 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
21ralrimiva 3149 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iunss 4936 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3sylibr 237 1 (𝜑 𝑥𝐴 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3883  ∪ ciun 4885 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-in 3890  df-ss 3900  df-iun 4887 This theorem is referenced by:  imasaddfnlem  16813  imasaddflem  16815  subdrgint  19596  meaiininclem  43293  smflim  43578  smfresal  43588  smfmullem4  43594
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