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Metamath Proof Explorer

Theorem elab3

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000) (Revised by AV, 16-Aug-2024)

Ref Expression
Hypotheses elab3.1 ⊢ ( 𝜓 → 𝐴 ∈ 𝑉 )
elab3.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion elab3 ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )

Proof

Step Hyp Ref Expression
1 elab3.1 ⊢ ( 𝜓 → 𝐴 ∈ 𝑉 )
2 elab3.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3 2 elab3g ⊢ ( ( 𝜓 → 𝐴 ∈ 𝑉 ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )
4 1 3 ax-mp ⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )