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

Theorem raldifeq 4422
Description: Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)
Hypotheses
Ref Expression
raldifeq.1 (𝜑𝐴𝐵)
raldifeq.2 (𝜑 → ∀𝑥 ∈ (𝐵𝐴)𝜓)
Assertion
Ref Expression
raldifeq (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem raldifeq
StepHypRef Expression
1 raldifeq.2 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐵𝐴)𝜓)
21biantrud 535 . . 3 (𝜑 → (∀𝑥𝐴 𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓)))
3 ralunb 4122 . . 3 (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓))
42, 3bitr4di 292 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓))
5 raldifeq.1 . . . 4 (𝜑𝐴𝐵)
6 undif 4413 . . . 4 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)
75, 6sylib 221 . . 3 (𝜑 → (𝐴 ∪ (𝐵𝐴)) = 𝐵)
87raleqdv 3340 . 2 (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ ∀𝑥𝐵 𝜓))
94, 8bitrd 282 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wral 3064  cdif 3880  cun 3881  wss 3883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255
This theorem is referenced by:  cantnfrescl  9321  rrxmet  24337  ntrneiel2  41421  ntrneik4w  41435
  Copyright terms: Public domain W3C validator