Theorem sbcrex

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005) (Revised by NM, 18-Aug-2018)

	Ref	Expression
Assertion	sbcrex	⊢ ( [ 𝐴 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfcv	⊢ Ⅎ 𝑦 𝐴
2		sbcrext	⊢ ( Ⅎ 𝑦 𝐴 → ( [ 𝐴 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 ) )
3	1 2	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 [ 𝐴 / 𝑥 ] 𝜑 )