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Theorem rspsbc 3331
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 2098 and spsbc 3265. See also rspsbca 3332 and rspcsbela 3773. (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 3020 . 2  |-  ( A. x  e.  B  ph  <->  A. y  e.  B  [ y  /  x ] ph )
2 dfsbcq2 3255 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rspcv 3131 . 2  |-  ( A  e.  B  ->  ( A. y  e.  B  [ y  /  x ] ph  ->  [. A  /  x ]. ph ) )
41, 3syl5bi 217 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   [wsb 1747    e. wcel 1826   A.wral 2732   [.wsbc 3252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-v 3036  df-sbc 3253
This theorem is referenced by:  rspsbca  3332  sbcth2  3336  rspcsbela  3773  riota5f  6182  riotass2  6184  fzrevral  11685  fprodcllemf  31757  rspsbc2  33641  truniALT  33648  rspsbc2VD  34001  truniALTVD  34025  trintALTVD  34027  trintALT  34028
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