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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 )