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Theorem spsbcdi 33093
 Description: A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
spsbcdi.1 𝐴 ∈ V
spsbcdi.2 (𝜑 → ∀𝑥𝜒)
spsbcdi.3 ([𝐴 / 𝑥]𝜒𝜓)
Assertion
Ref Expression
spsbcdi (𝜑𝜓)

Proof of Theorem spsbcdi
StepHypRef Expression
1 spsbcdi.1 . . . 4 𝐴 ∈ V
21a1i 11 . . 3 (𝜑𝐴 ∈ V)
3 spsbcdi.2 . . 3 (𝜑 → ∀𝑥𝜒)
42, 3spsbcd 3416 . 2 (𝜑[𝐴 / 𝑥]𝜒)
5 spsbcdi.3 . 2 ([𝐴 / 𝑥]𝜒𝜓)
64, 5sylib 207 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  Vcvv 3173  [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-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by: (None)
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