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Theorem iunssd 4959
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 3105 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iunss 4954 . 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 2110  wral 3061  wss 3866   ciun 4904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-v 3410  df-in 3873  df-ss 3883  df-iun 4906
This theorem is referenced by:  imasaddfnlem  17033  imasaddflem  17035  subdrgint  19847  meaiininclem  43699  smflim  43984  smfresal  43994  smfmullem4  44000
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