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Theorem spsbcdi 32876
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 3415 . 2 (𝜑[𝐴 / 𝑥]𝜒)
5 spsbcdi.3 . 2 ([𝐴 / 𝑥]𝜒𝜓)
64, 5sylib 206 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wcel 1976  Vcvv 3172  [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-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174  df-sbc 3402
This theorem is referenced by: (None)
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