Theorem cbvexsv

Description: A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	cbvexsv	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		cbvrexsv	⊢ ( ∃ 𝑥 ∈ V 𝜑 ↔ ∃ 𝑦 ∈ V [ 𝑦 / 𝑥 ] 𝜑 )
2		rexv	⊢ ( ∃ 𝑥 ∈ V 𝜑 ↔ ∃ 𝑥 𝜑 )
3		rexv	⊢ ( ∃ 𝑦 ∈ V [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
4	1 2 3	3bitr3i	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )