Theorem rspsbc 3072

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 1789 and spsbc 3001. See also rspsbca 3073 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 2745	. 2 |- ( A. x e. B ph <-> A. y e. B [ y / x ] ph )
2		dfsbcq2 2992	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	rspcv 2864	. 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 1776   e. wcel 2167   A.wral 2475   [.wsbc 2989

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-sbc 2990

This theorem is referenced by:  rspsbca  3073  sbcth2  3077  rspcsbela  3144  riota5f  5902  riotass2  5904  fzrevral  10180  fprodcllemf  11778  ctiunctlemf  12655