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Theorem rspsbc 3069
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 1964 and spsbc 3003. See also rspsbca 3070 and rspcsbela 3140. (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 2775 . 2  |-  ( A. x  e.  B  ph  <->  A. y  e.  B  [ y  /  x ] ph )
2 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rspcv 2880 . 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 1629    e. wcel 1684   A.wral 2543   [.wsbc 2991
This theorem is referenced by:  rspsbca  3070  sbcth2  3074  rspcsbela  3140  riota5f  6329  riotass2  6332  fzrevral  10866  rspsbc2  28297  truniALT  28305  rspsbc2VD  28631  truniALTVD  28654  trintALTVD  28656  trintALT  28657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992
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