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 )