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Metamath Proof Explorer

Theorem eluniab

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994) (Revised by Mario Carneiro, 14-Nov-2016)

Ref Expression
Assertion eluniab ( 𝐴 ∈ ∪ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝜑 ) )

Proof

Step Hyp Ref Expression
1 eluni ⊢ ( 𝐴 ∈ ∪ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑦 ( 𝐴 ∈ 𝑦 ∧ 𝑦 ∈ { 𝑥 ∣ 𝜑 } ) )
2 nfv ⊢ Ⅎ 𝑥 𝐴 ∈ 𝑦
3 nfsab1 ⊢ Ⅎ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜑 }
4 2 3 nfan ⊢ Ⅎ 𝑥 ( 𝐴 ∈ 𝑦 ∧ 𝑦 ∈ { 𝑥 ∣ 𝜑 } )
5 nfv ⊢ Ⅎ 𝑦 ( 𝐴 ∈ 𝑥 ∧ 𝜑 )
6 eleq2w ⊢ ( 𝑦 = 𝑥 → ( 𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥 ) )
7 eleq1w ⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) )
8 abid ⊢ ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 )
9 7 8 bitrdi ⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 ) )
10 6 9 anbi12d ⊢ ( 𝑦 = 𝑥 → ( ( 𝐴 ∈ 𝑦 ∧ 𝑦 ∈ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝐴 ∈ 𝑥 ∧ 𝜑 ) ) )
11 4 5 10 cbvexv1 ⊢ ( ∃ 𝑦 ( 𝐴 ∈ 𝑦 ∧ 𝑦 ∈ { 𝑥 ∣ 𝜑 } ) ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝜑 ) )
12 1 11 bitri ⊢ ( 𝐴 ∈ ∪ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝐴 ∈ 𝑥 ∧ 𝜑 ) )