Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralss Structured version   Visualization version   GIF version

Theorem ralss 4013
 Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
ralss (𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralss
StepHypRef Expression
1 ssel 3937 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21pm4.71rd 566 . . . 4 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
32imbi1d 345 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐵𝑥𝐴) → 𝜑)))
4 impexp 454 . . 3 (((𝑥𝐵𝑥𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥𝐴𝜑)))
53, 4syl6bb 290 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵 → (𝑥𝐴𝜑))))
65ralbidv2 3183 1 (𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2115  ∀wral 3126   ⊆ wss 3910 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-v 3473  df-in 3917  df-ss 3927 This theorem is referenced by:  acsfn  16908  acsfn1  16910  acsfn2  16912  acsfn1p  19553  mdetunilem9  21204  ntrneik3  40581  ntrneix3  40582  ntrneik13  40583  ntrneix13  40584
 Copyright terms: Public domain W3C validator