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Metamath Proof Explorer

Theorem rabexd

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 . (Contributed by AV, 16-Jul-2019)

Ref Expression
Hypotheses rabexd.1 ⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜓 }
rabexd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion rabexd ( 𝜑 → 𝐵 ∈ V )

Proof

Step Hyp Ref Expression
1 rabexd.1 ⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜓 }
2 rabexd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3 rabexg ⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ∈ V )
4 2 3 syl ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } ∈ V )
5 1 4 eqeltrid ⊢ ( 𝜑 → 𝐵 ∈ V )