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Theorem iunssd 4840
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 3133 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iunss 4835 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3sylibr 226 1 (𝜑 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2050  wral 3089  wss 3830   ciun 4792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-in 3837  df-ss 3844  df-iun 4794
This theorem is referenced by:  imasaddfnlem  16657  imasaddflem  16659  subdrgint  19304  meaiininclem  42197  smflim  42482  smfresal  42492  smfmullem4  42498
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