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Theorem sbexi 34248
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 2183 . . 3 𝑥𝑥𝜑
32sbcgf 3651 . 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 1852  wcel 2145  Vcvv 3351  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by: (None)
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