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Theorem ssralv 2998
Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
Assertion
Ref Expression
ssralv (AB → (x B φx A φ))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem ssralv
StepHypRef Expression
1 ssel 2933 . . 3 (AB → (x Ax B))
21imim1d 69 . 2 (AB → ((x Bφ) → (x Aφ)))
32ralimdv2 2383 1 (AB → (x B φx A φ))
Colors of variables: wff set class
Syntax hints:  wi 4   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:  iinss1  3660  poss  4026  sess2  4060  trssord  4083  funco  4883  funimaexglem  4925  isores3  5398  isoini2  5401  smores  5848  smores2  5850  tfrlem5  5871  peano5nni  7678
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