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Theorem raldifeq 4011
 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 527 . . 3 (𝜑 → (∀𝑥𝐴 𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓)))
3 ralunb 3756 . . 3 (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓))
42, 3syl6bbr 277 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓))
5 raldifeq.1 . . . 4 (𝜑𝐴𝐵)
6 undif 4001 . . . 4 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)
75, 6sylib 207 . . 3 (𝜑 → (𝐴 ∪ (𝐵𝐴)) = 𝐵)
87raleqdv 3121 . 2 (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ ∀𝑥𝐵 𝜓))
94, 8bitrd 267 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∀wral 2896   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  cantnfrescl  8456  rrxmet  22999  ntrneiel2  37404  ntrneik4w  37418
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