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Theorem rspsbca 3860
Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
Assertion
Ref Expression
rspsbca ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → [𝐴 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspsbca
StepHypRef Expression
1 rspsbc 3859 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
21imp 407 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-sbc 3770
This theorem is referenced by:  fprodmodd  15339  telgsums  19042  iccelpart  43470
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