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Theorem rspsbc 3124
 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 2024 and spsbc 3058. See also rspsbca 3125 and rspcsbela 3195. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
rspsbc (A B → (x B φ → [̣A / xφ))
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 2846 . 2 (x B φy B [y / x]φ)
2 dfsbcq2 3049 . . 3 (y = A → ([y / x]φ ↔ [̣A / xφ))
32rspcv 2951 . 2 (A B → (y B [y / x]φ → [̣A / xφ))
41, 3syl5bi 208 1 (A B → (x B φ → [̣A / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 1648   ∈ wcel 1710  ∀wral 2614  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047 This theorem is referenced by:  rspsbca  3125  sbcth2  3129  rspcsbela  3195
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