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Theorem ra4sbc 2999
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 1896 and a4sbc 2933. See also ra4sbca 3000 and ra4csbela 3068. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
ra4sbc  |-  ( 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 ra4sbc
StepHypRef Expression
1 cbvralsv 2714 . 2  |-  ( A. x  e.  B  ph  <->  A. y  e.  B  [ y  /  x ] ph )
2 dfsbcq2 2924 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rcla4v 2817 . 2  |-  ( A  e.  B  ->  ( A. y  e.  B  [ y  /  x ] ph  ->  [. A  /  x ]. ph ) )
41, 3syl5bi 210 1  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   [wsb 1882   A.wral 2509   [.wsbc 2921
This theorem is referenced by:  ra4sbca  3000  sbcth2  3004  ra4csbela  3068  riota5f  6215  riotass2  6218  fzrevral  10744  ra4sbc2  26990  truniALT  26998  ra4sbc2VD  27321  truniALTVD  27344  trintALTVD  27346  trintALT  27347
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729  df-sbc 2922
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