Theorem rspsbca 3090

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 3089	. 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 2178   A.wral 2486   [.wsbc 3005

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-sbc 3006

This theorem is referenced by:  fprodmodd  12067