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Theorem sbexi 38073
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 2151 . 2 𝑥𝑥𝜑
31, 2sbcgfi 3885 1 ([𝐴 / 𝑥]𝑥𝜑 ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1777  wcel 2108  Vcvv 3488  [wsbc 3804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805
This theorem is referenced by: (None)
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