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Metamath Proof Explorer

Theorem bnj1230

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypothesis bnj1230.1 ⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
Assertion bnj1230 ( 𝑦 ∈ 𝐵 → ∀ 𝑥 𝑦 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 bnj1230.1 ⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
2 nfrab1 ⊢ Ⅎ 𝑥 { 𝑥 ∈ 𝐴 ∣ 𝜑 }
3 1 2 nfcxfr ⊢ Ⅎ 𝑥 𝐵
4 3 nfcrii ⊢ ( 𝑦 ∈ 𝐵 → ∀ 𝑥 𝑦 ∈ 𝐵 )