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Metamath Proof Explorer

Theorem elrab2

Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006)

Ref Expression
Hypotheses elrab2.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
elrab2.2 ⊢ 𝐶 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
Assertion elrab2 ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 elrab2.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2 elrab2.2 ⊢ 𝐶 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
3 2 eleq2i ⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } )
4 1 elrab ⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )
5 3 4 bitri ⊢ ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )