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Theorem expanduniss 43517
Description: Expand 𝐴𝐵 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 4943 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 dfss2 3968 . . 3 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
32expandral 43514 . 2 (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
41, 3bitri 275 1 ( 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538  wcel 2105  wral 3060  wss 3948   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909
This theorem is referenced by:  ismnuprim  43518
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