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Theorem rspsbc 3031
Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1762 and spsbc 2960. See also rspsbca 3032 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
rspsbc  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem rspsbc
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 2706 . 2  |-  ( A. x  e.  B  ph  <->  A. y  e.  B  [ y  /  x ] ph )
2 dfsbcq2 2952 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rspcv 2824 . 2  |-  ( A  e.  B  ->  ( A. y  e.  B  [ y  /  x ] ph  ->  [. A  /  x ]. ph ) )
41, 3syl5bi 151 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [wsb 1749    e. wcel 2135   A.wral 2442   [.wsbc 2949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2726  df-sbc 2950
This theorem is referenced by:  rspsbca  3032  sbcth2  3036  rspcsbela  3102  riota5f  5819  riotass2  5821  fzrevral  10034  fprodcllemf  11548  ctiunctlemf  12365
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