MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspsbc Structured version   Visualization version   GIF version

Theorem rspsbc 3713
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 2469 and spsbc 3646. See also rspsbca 3714 and rspcsbela 4202. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
rspsbc (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspsbc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cbvralsv 3365 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑦 / 𝑥]𝜑)
2 dfsbcq2 3636 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rspcv 3493 . 2 (𝐴𝐵 → (∀𝑦𝐵 [𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
41, 3syl5bi 234 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2064  wcel 2157  wral 3089  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-sbc 3634
This theorem is referenced by:  rspsbca  3714  sbcth2  3718  rspcsbela  4202  riota5f  6864  riotass2  6866  fzrevral  12679  fprodcllemf  15025  rspsbc2  39520  truniALT  39527  rspsbc2VD  39851  truniALTVD  39874  trintALTVD  39876  trintALT  39877
  Copyright terms: Public domain W3C validator