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Metamath Proof Explorer

Theorem opelopab2a

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013)

Ref Expression
Hypothesis opelopabga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion opelopab2a ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 opelopabga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
2 eleq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
3 eleq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷 ) )
4 2 3 bi2anan9 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) )
5 4 1 anbi12d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) ↔ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝜓 ) ) )
6 5 opelopabga ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝜓 ) ) )
7 6 bianabs ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ 𝜓 ) )