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Theorem rspsbc 3483
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 2340 and spsbc 3414. See also rspsbca 3484 and rspcsbela 3957. (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 3157 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑦 / 𝑥]𝜑)
2 dfsbcq2 3404 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rspcv 3277 . 2 (𝐴𝐵 → (∀𝑦𝐵 [𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
41, 3syl5bi 230 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1866  wcel 1976  wral 2895  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174  df-sbc 3402
This theorem is referenced by:  rspsbca  3484  sbcth2  3488  rspcsbela  3957  riota5f  6513  riotass2  6515  fzrevral  12252  fprodcllemf  14476  rspsbc2  37589  truniALT  37596  rspsbc2VD  37936  truniALTVD  37960  trintALTVD  37962  trintALT  37963
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