Theorem bj-sbf3 33401

Description: Substitution has no effect on a bound variabe (existential quantifier case); see sbf2 2457. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-sbf3	⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem bj-sbf3

Step	Hyp	Ref	Expression
1		nfe1 2143	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	sbf 2455	1 ⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 198  ∃wex 1823  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162  ax-13 2333

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by: (None)