Theorem sbcopeq1a

Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a that avoids the existential quantifiers of copsexg ). (Contributed by NM, 19-Aug-2006) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Assertion	sbcopeq1a	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( [ ( 1st ‘ 𝐴 ) / 𝑥 ] [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2		vex	⊢ 𝑦 ∈ V
3	1 2	op2ndd	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 2nd ‘ 𝐴 ) = 𝑦 )
4	3	eqcomd	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → 𝑦 = ( 2nd ‘ 𝐴 ) )
5		sbceq1a	⊢ ( 𝑦 = ( 2nd ‘ 𝐴 ) → ( 𝜑 ↔ [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ) )
6	4 5	syl	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ) )
7	1 2	op1std	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 1st ‘ 𝐴 ) = 𝑥 )
8	7	eqcomd	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → 𝑥 = ( 1st ‘ 𝐴 ) )
9		sbceq1a	⊢ ( 𝑥 = ( 1st ‘ 𝐴 ) → ( [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ↔ [ ( 1st ‘ 𝐴 ) / 𝑥 ] [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ) )
10	8 9	syl	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ↔ [ ( 1st ‘ 𝐴 ) / 𝑥 ] [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ) )
11	6 10	bitr2d	⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( [ ( 1st ‘ 𝐴 ) / 𝑥 ] [ ( 2nd ‘ 𝐴 ) / 𝑦 ] 𝜑 ↔ 𝜑 ) )