Theorem rspsbca 3073

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 3072	. 2 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )
2	1	imp 124	1 |- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   A.wral 2475   [.wsbc 2989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-sbc 2990

This theorem is referenced by:  fprodmodd  11806