Theorem rspsbc 3358

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 2195 and spsbc 3292. See also rspsbca 3359 and rspcsbela 3807. (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 3042	. 2 |- ( A. x e. B ph <-> A. y e. B [ y / x ] ph )
2		dfsbcq2 3282	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	rspcv 3158	. 2 |- ( A e. B -> ( A. y e. B [ y / x ] ph -> [. A / x ]. ph ) )
4	1, 3	syl5bi 225	1 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   [wsb 1808   e. wcel 1898   A.wral 2749   [.wsbc 3279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-v 3059  df-sbc 3280

This theorem is referenced by:  rspsbca  3359  sbcth2  3363  rspcsbela  3807  riota5f  6301  riotass2  6303  fzrevral  11908  fprodcllemf  14061  rspsbc2  36939  truniALT  36946  rspsbc2VD  37291  truniALTVD  37315  trintALTVD  37317  trintALT  37318