Theorem sbexi 37861

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 2143	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
3	1, 2	sbcgfi 3869	1 ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1774   ∈ wcel 2100  Vcvv 3472  [wsbc 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-12 2170  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-sbc 3789

This theorem is referenced by: (None)