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Metamath Proof Explorer

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