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Theorem rspsbca 2964
Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
Assertion
Ref Expression
rspsbca  |-  ( ( 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 rspsbca
StepHypRef Expression
1 rspsbc 2963 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
21imp 123 1  |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   A.wral 2393   [.wsbc 2882
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-sbc 2883
This theorem is referenced by: (None)
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