Theorem bnj1317

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

	Ref	Expression
Hypothesis	bnj1317.1	⊢ 𝐴 = { 𝑥 ∣ 𝜑 }
Assertion	bnj1317	⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		bnj1317.1	⊢ 𝐴 = { 𝑥 ∣ 𝜑 }
2		hbab1	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } → ∀ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜑 } )
3	1 2	hbxfreq	⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )