Theorem altopthbg

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012)

	Ref	Expression
Assertion	altopthbg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ) → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		altopthsn	⊢ ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) )
2		sneqbg	⊢ ( 𝐴 ∈ 𝑉 → ( { 𝐴 } = { 𝐶 } ↔ 𝐴 = 𝐶 ) )
3		sneqbg	⊢ ( 𝐷 ∈ 𝑊 → ( { 𝐷 } = { 𝐵 } ↔ 𝐷 = 𝐵 ) )
4		eqcom	⊢ ( { 𝐵 } = { 𝐷 } ↔ { 𝐷 } = { 𝐵 } )
5		eqcom	⊢ ( 𝐵 = 𝐷 ↔ 𝐷 = 𝐵 )
6	3 4 5	3bitr4g	⊢ ( 𝐷 ∈ 𝑊 → ( { 𝐵 } = { 𝐷 } ↔ 𝐵 = 𝐷 ) )
7	2 6	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ) → ( ( { 𝐴 } = { 𝐶 } ∧ { 𝐵 } = { 𝐷 } ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )
8	1 7	bitrid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ) → ( ⟪ 𝐴 , 𝐵 ⟫ = ⟪ 𝐶 , 𝐷 ⟫ ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) )