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

Theorem rspsbc 3551
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 2381 and spsbc 3481. See also rspsbca 3552 and rspcsbela 4039. (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 3212 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑦 / 𝑥]𝜑)
2 dfsbcq2 3471 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rspcv 3336 . 2 (𝐴𝐵 → (∀𝑦𝐵 [𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
41, 3syl5bi 232 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1937  wcel 2030  wral 2941  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-sbc 3469
This theorem is referenced by:  rspsbca  3552  sbcth2  3556  rspcsbela  4039  riota5f  6676  riotass2  6678  fzrevral  12463  fprodcllemf  14732  rspsbc2  39061  truniALT  39068  rspsbc2VD  39404  truniALTVD  39428  trintALTVD  39430  trintALT  39431
  Copyright terms: Public domain W3C validator