Theorem bj-sbf3 37506

Description: Substitution has no effect on a bound variable (existential quantifier case); see sbf2 2310. (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 2188	. 2 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	sbf 2309	1 ⊢ ([𝑦 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by: (None)