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Metamath Proof Explorer

Theorem elintab

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993)

Ref Expression
Hypothesis elintab.ex ⊢ 𝐴 ∈ V
Assertion elintab ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) )

Proof

Step Hyp Ref Expression
1 elintab.ex ⊢ 𝐴 ∈ V
2 elintabg ⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )
3 1 2 ax-mp ⊢ ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) )