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Theorem expanduniss 40994
 Description: Expand ∪ 𝐴 ⊆ 𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Assertion
Ref Expression
expanduniss ( 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem expanduniss
StepHypRef Expression
1 unissb 4835 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 dfss2 3904 . . 3 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
32expandral 40991 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
41, 3bitri 278 1 ( 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  ∪ cuni 4803 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 2114  ax-9 2122  ax-11 2159  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804 This theorem is referenced by:  ismnuprim  40995
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