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Theorem iunssd 5053
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 3143 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iunss 5048 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3sylibr 233 1 (𝜑 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wral 3058  wss 3947   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-v 3473  df-in 3954  df-ss 3964  df-iun 4998
This theorem is referenced by:  imasaddfnlem  17509  imasaddflem  17511  subdrgint  20690  precsexlem10  28113  oacl2g  42759  omcl2  42762  ofoaf  42784  onsucunifi  42799  meaiininclem  45874  smflim  46165  smfresal  46176  smfmullem4  46182
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