Metamath Proof Explorer


Theorem sb8e

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8ev . (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.)

Ref Expression
Hypothesis sb8.1 𝑦 𝜑
Assertion sb8e ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 sb8.1 𝑦 𝜑
2 1 nfs1 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3 sbequ12 ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4 1 2 3 cbvex ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )