Theorem sb8e 2547

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2401. For a version requiring disjoint variables, but fewer axioms, see sb8ef 2384. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb8.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8e	⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb8.1	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfs1 2517	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2286	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvex 2428	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  Ⅎwnf 1816  [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 2178  ax-11 2194  ax-12 2213  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  2sb8e  2559  sb8mo  2626  bnj985  35464  exlimddvfi  38871  pm11.58  45215