Theorem rspsbca 2996

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

Step	Hyp	Ref	Expression
1		rspsbc 2995	. 2 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )
2	1	imp 123	1 |- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1481   A.wral 2417   [.wsbc 2913

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-sbc 2914

This theorem is referenced by: (None)