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Theorem rspsbc 3866
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 2063 and spsbc 3783. See also rspsbca 3867 and rspcsbela 4428. (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 cbvralsvw 3306 . 2 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑦 / 𝑥]𝜑)
2 dfsbcq2 3773 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32rspcv 3600 . 2 (𝐴𝐵 → (∀𝑦𝐵 [𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
41, 3biimtrid 241 1 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2059  wcel 2098  wral 3053  [wsbc 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-sbc 3771
This theorem is referenced by:  rspsbca  3867  sbcth2  3871  rspcsbela  4428  riota5f  7387  riotass2  7389  fzrevral  13587  fprodcllemf  15904  rspsbc2  43845  truniALT  43852  rspsbc2VD  44166  truniALTVD  44189  trintALTVD  44191  trintALT  44192
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