Theorem rspsbc 3033

Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1763 and spsbc 2962. See also rspsbca 3034 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspsbc	|- ( 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 rspsbc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2708	. 2 |- ( A. x e. B ph <-> A. y e. B [ y / x ] ph )
2		dfsbcq2 2954	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	rspcv 2826	. 2 |- ( A e. B -> ( A. y e. B [ y / x ] ph -> [. A / x ]. ph ) )
4	1, 3	syl5bi 151	1 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   [wsb 1750   e. wcel 2136   A.wral 2444   [.wsbc 2951

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-sbc 2952

This theorem is referenced by:  rspsbca  3034  sbcth2  3038  rspcsbela  3104  riota5f  5822  riotass2  5824  fzrevral  10040  fprodcllemf  11554  ctiunctlemf  12371