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Theorem sbexi 32316
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  e. 
_V
Assertion
Ref Expression
sbexi  |-  ( [. A  /  x ]. E. x ph  <->  E. x ph )

Proof of Theorem sbexi
StepHypRef Expression
1 sbexi.1 . 2  |-  A  e. 
_V
2 nfe1 1894 . . 3  |-  F/ x E. x ph
32sbcgf 3363 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E. x ph  <->  E. x ph ) )
41, 3ax-mp 5 1  |-  ( [. A  /  x ]. E. x ph  <->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   E.wex 1657    e. wcel 1872   _Vcvv 3080   [.wsbc 3299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3082  df-sbc 3300
This theorem is referenced by: (None)
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