Theorem opelopab4

Description: Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab . (Contributed by Alan Sare, 8-Feb-2014) (Revised by Mario Carneiro, 6-May-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	opelopab4	⊢ ( ⟨ 𝑢 , 𝑣 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		elopab	⊢ ( ⟨ 𝑢 , 𝑣 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑢 , 𝑣 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
2		vex	⊢ 𝑥 ∈ V
3		vex	⊢ 𝑦 ∈ V
4	2 3	opth	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑢 , 𝑣 ⟩ ↔ ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) )
5		eqcom	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑢 , 𝑣 ⟩ ↔ ⟨ 𝑢 , 𝑣 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
6	4 5	bitr3i	⊢ ( ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ↔ ⟨ 𝑢 , 𝑣 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
7	6	anbi1i	⊢ ( ( ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ∧ 𝜑 ) ↔ ( ⟨ 𝑢 , 𝑣 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
8	7	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑢 , 𝑣 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
9	1 8	bitr4i	⊢ ( ⟨ 𝑢 , 𝑣 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑢 ∧ 𝑦 = 𝑣 ) ∧ 𝜑 ) )