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Theorem ralss 3332
 Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
ralss (A B → (x A φx B (x Aφ)))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem ralss
StepHypRef Expression
1 ssel 3267 . . . . 5 (A B → (x Ax B))
21pm4.71rd 616 . . . 4 (A B → (x A ↔ (x B x A)))
32imbi1d 308 . . 3 (A B → ((x Aφ) ↔ ((x B x A) → φ)))
4 impexp 433 . . 3 (((x B x A) → φ) ↔ (x B → (x Aφ)))
53, 4syl6bb 252 . 2 (A B → ((x Aφ) ↔ (x B → (x Aφ))))
65ralbidv2 2636 1 (A B → (x A φx B (x Aφ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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