Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  expanduniss Structured version   Visualization version   GIF version

Theorem expanduniss 44248
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 4946 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
2 df-ss 3980 . . 3 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
32expandral 44245 . 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 206  wal 1533  wcel 2104  wral 3057  wss 3963   cuni 4914
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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-v 3479  df-ss 3980  df-uni 4915
This theorem is referenced by:  ismnuprim  44249
  Copyright terms: Public domain W3C validator