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Theorem rspsbc 3082
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 1977 and spsbc 3016. See also rspsbca 3083 and rspcsbela 3153. (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 2788 . 2  |-  ( A. x  e.  B  ph  <->  A. y  e.  B  [ y  /  x ] ph )
2 dfsbcq2 3007 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rspcv 2893 . 2  |-  ( A  e.  B  ->  ( A. y  e.  B  [ y  /  x ] ph  ->  [. A  /  x ]. ph ) )
41, 3syl5bi 208 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [wsb 1638    e. wcel 1696   A.wral 2556   [.wsbc 3004
This theorem is referenced by:  rspsbca  3083  sbcth2  3087  rspcsbela  3153  riota5f  6345  riotass2  6348  fzrevral  10882  rspsbc2  28596  truniALT  28604  rspsbc2VD  28947  truniALTVD  28970  trintALTVD  28972  trintALT  28973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-sbc 3005
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