Theorem sbexi 38152

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 2153	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
3	1, 2	sbcgfi 3815	1 ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 206  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  [wsbc 3741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3742

This theorem is referenced by: (None)