Theorem rspsbc 3080

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 1797 and spsbc 3009. See also rspsbca 3081 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 2753	. 2 |- ( A. x e. B ph <-> A. y e. B [ y / x ] ph )
2		dfsbcq2 3000	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	rspcv 2872	. 2 |- ( A e. B -> ( A. y e. B [ y / x ] ph -> [. A / x ]. ph ) )
4	1, 3	biimtrid 152	1 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   [wsb 1784   e. wcel 2175   A.wral 2483   [.wsbc 2997

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-sbc 2998

This theorem is referenced by:  rspsbca  3081  sbcth2  3085  rspcsbela  3152  riota5f  5923  riotass2  5925  fzrevral  10226  fprodcllemf  11866  ctiunctlemf  12751