Theorem sbexi 36686

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

Step	Hyp	Ref	Expression
1		sbexi.1	. 2 ⊢ 𝐴 ∈ V
2		nfe1 2147	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
3	1, 2	sbcgfi 3845	1 ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1781   ∈ wcel 2106  Vcvv 3466  [wsbc 3764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-sbc 3765

This theorem is referenced by: (None)