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Theorem raldifeq 4488
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 530 . . 3 (𝜑 → (∀𝑥𝐴 𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓)))
3 ralunb 4189 . . 3 (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓))
42, 3bitr4di 288 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓))
5 raldifeq.1 . . . 4 (𝜑𝐴𝐵)
6 undif 4476 . . . 4 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)
75, 6sylib 217 . . 3 (𝜑 → (𝐴 ∪ (𝐵𝐴)) = 𝐵)
87raleqdv 3315 . 2 (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ ∀𝑥𝐵 𝜓))
94, 8bitrd 278 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wral 3051  cdif 3943  cun 3944  wss 3946
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323
This theorem is referenced by:  cantnfrescl  9709  rrxmet  25421  ntrneiel2  43787  ntrneik4w  43801
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