Theorem rspsbca 3519

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 3518	. 2 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )
2	1	imp 445	1 |- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by:  fprodmodd  14728  telgsums  18390  iccelpart  41369