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Theorem raldifeq 4203
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 529 . . 3 (𝜑 → (∀𝑥𝐴 𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓)))
3 ralunb 3937 . . 3 (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ (∀𝑥𝐴 𝜓 ∧ ∀𝑥 ∈ (𝐵𝐴)𝜓))
42, 3syl6bbr 278 . 2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓))
5 raldifeq.1 . . . 4 (𝜑𝐴𝐵)
6 undif 4193 . . . 4 (𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)
75, 6sylib 208 . . 3 (𝜑 → (𝐴 ∪ (𝐵𝐴)) = 𝐵)
87raleqdv 3283 . 2 (𝜑 → (∀𝑥 ∈ (𝐴 ∪ (𝐵𝐴))𝜓 ↔ ∀𝑥𝐵 𝜓))
94, 8bitrd 268 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wral 3050  cdif 3712  cun 3713  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059
This theorem is referenced by:  cantnfrescl  8746  rrxmet  23391  ntrneiel2  38886  ntrneik4w  38900
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