Theorem sb8e 2518

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2372. For a version requiring disjoint variables, but fewer axioms, see sb8ef 2355. (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 2488	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2254	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvex 2399	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  ∃wex 1780  Ⅎwnf 1784  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  2sb8e  2530  sb8mo  2596  bnj985  34966  exlimddvfi  38170  pm11.58  44431