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Theorem rspsbca 3146
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 3145 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  ph 
->  [. A  /  x ]. ph ) )
21imp 418 1  |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   A.wral 2619   [.wsbc 3067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-v 2866  df-sbc 3068
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