Theorem rspsbc 3417

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 2094 and spsbc 3340. See also rspsbca 3418 and rspcsbela 3853. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspsbc

Distinct variable group:   ,

Proof of Theorem rspsbc

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 3095	. 2
2		dfsbcq2 3330	. . 3
3	2	rspcv 3206	. 2
4	1, 3	syl5bi 217	1

Syntax hints:  ->wi 4  [wsb 1739  e.wcel 1818  A.wral 2807  [.wsbc 3327

This theorem is referenced by:  rspsbca  3418  sbcth2  3422  rspcsbela  3853  riota5f  6282  riotass2  6284  fzrevral  11792  fprodcllemf  31591  rspsbc2  33305  truniALT  33312  rspsbc2VD  33655  truniALTVD  33678  trintALTVD  33680  trintALT  33681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328