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Theorem ralss 3076
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 3008 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21pm4.71rd 386 . . . 4 (𝐴𝐵 → (𝑥𝐴 ↔ (𝑥𝐵𝑥𝐴)))
32imbi1d 229 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ ((𝑥𝐵𝑥𝐴) → 𝜑)))
4 impexp 259 . . 3 (((𝑥𝐵𝑥𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥𝐴𝜑)))
53, 4syl6bb 194 . 2 (𝐴𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵 → (𝑥𝐴𝜑))))
65ralbidv2 2378 1 (𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1436  wral 2355  wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-ral 2360  df-in 2994  df-ss 3001
This theorem is referenced by: (None)
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