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Theorem ralss 3000
 Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
ralss (AB → (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 2933 . . . . 5 (AB → (x Ax B))
21pm4.71rd 374 . . . 4 (AB → (x A ↔ (x B x A)))
32imbi1d 220 . . 3 (AB → ((x Aφ) ↔ ((x B x A) → φ)))
4 impexp 250 . . 3 (((x B x A) → φ) ↔ (x B → (x Aφ)))
53, 4syl6bb 185 . 2 (AB → ((x Aφ) ↔ (x B → (x Aφ))))
65ralbidv2 2322 1 (AB → (x A φx B (x Aφ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925 This theorem is referenced by: (None)
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