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Theorem rspsbc 3484
 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 2341 and spsbc 3415. See also rspsbca 3485 and rspcsbela 3958. (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 3158 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑦 / 𝑥]𝜑)
2 dfsbcq2 3405 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rspcv 3278 . 2 (𝐴𝐵 → (∀𝑦𝐵 [𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
41, 3syl5bi 231 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 1867   ∈ wcel 1977  ∀wral 2896  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-sbc 3403 This theorem is referenced by:  rspsbca  3485  sbcth2  3489  rspcsbela  3958  riota5f  6535  riotass2  6537  fzrevral  12294  fprodcllemf  14527  rspsbc2  37765  truniALT  37772  rspsbc2VD  38112  truniALTVD  38136  trintALTVD  38138  trintALT  38139
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