This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )