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Theorem sbexi 34456
 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 2201 . . 3 𝑥𝑥𝜑
32sbcgf 3726 . 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 198  ∃wex 1878   ∈ wcel 2164  Vcvv 3414  [wsbc 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663 This theorem is referenced by: (None)
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