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Theorem sbexi 33887
 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 2025 . . 3 𝑥𝑥𝜑
32sbcgf 3495 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝜑 ↔ ∃𝑥𝜑))
41, 3ax-mp 5 1 ([𝐴 / 𝑥]𝑥𝜑 ↔ ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∃wex 1702   ∈ wcel 1988  Vcvv 3195  [wsbc 3429 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197  df-sbc 3430 This theorem is referenced by: (None)
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