Theorem altopth2

Description: Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012)

	Ref	Expression
Assertion	altopth2	⊢ ( 𝐵 ∈ 𝑉 → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ → 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		altopthsn	⊢ ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) )
2		sneqrg	⊢ ( 𝐵 ∈ 𝑉 → ( { 𝐵 } = { 𝐷 } → 𝐵 = 𝐷 ) )
3	2	adantld	⊢ ( 𝐵 ∈ 𝑉 → ( ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) → 𝐵 = 𝐷 ) )
4	1 3	biimtrid	⊢ ( 𝐵 ∈ 𝑉 → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ → 𝐵 = 𝐷 ) )