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Theorem spsbcdi 35461
 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 3771 . 2 (𝜑[𝐴 / 𝑥]𝜒)
5 spsbcdi.3 . 2 ([𝐴 / 𝑥]𝜒𝜓)
64, 5sylib 221 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2115  Vcvv 3479  [wsbc 3757 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-sbc 3758 This theorem is referenced by: (None)
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