Theorem sbexi 38487

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 2161	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
3	1, 2	sbcgfi 3803	1 ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 207  ∃wex 1786   ∈ wcel 2119  Vcvv 3432  [wsbc 3730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-sbc 3731

This theorem is referenced by: (None)