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Theorem ralss 3774
 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 3703 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21pm4.71rd 670 . . . 4 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
32imbi1d 330 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐵𝑥𝐴) → 𝜑)))
4 impexp 461 . . 3 (((𝑥𝐵𝑥𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥𝐴𝜑)))
53, 4syl6bb 276 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵 → (𝑥𝐴𝜑))))
65ralbidv2 3086 1 (𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2103  ∀wral 3014   ⊆ wss 3680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-ral 3019  df-in 3687  df-ss 3694 This theorem is referenced by:  acsfn  16442  acsfn1  16444  acsfn2  16446  mdetunilem9  20549  acsfn1p  38188  ntrneik3  38813  ntrneix3  38814  ntrneik13  38815  ntrneix13  38816
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