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Theorem sbexi 35390
Description: Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypothesis
Ref Expression
sbexi.1 𝐴 ∈ V
Assertion
Ref Expression
sbexi ([𝐴 / 𝑥]𝑥𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem sbexi
StepHypRef Expression
1 sbexi.1 . 2 𝐴 ∈ V
2 nfe1 2150 . 2 𝑥𝑥𝜑
31, 2sbcgfi 3847 1 ([𝐴 / 𝑥]𝑥𝜑 ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wex 1776  wcel 2110  Vcvv 3494  [wsbc 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3772
This theorem is referenced by: (None)
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