Theorem rspsbc 3115

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 1823 and spsbc 3043. See also rspsbca 3116 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 2783	. 2 |- ( A. x e. B ph <-> A. y e. B [ y / x ] ph )
2		dfsbcq2 3034	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	rspcv 2906	. 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 1810   e. wcel 2202   A.wral 2510   [.wsbc 3031

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-sbc 3032

This theorem is referenced by:  rspsbca  3116  sbcth2  3120  rspcsbela  3187  riota5f  5997  riotass2  5999  fzrevral  10339  fprodcllemf  12173  ctiunctlemf  13058