Theorem ineq1

Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993) (Proof shortened by SN, 20-Sep-2023)

	Ref	Expression
Assertion	ineq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rabeq	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐶 } = { 𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐶 } )
2		dfin5	⊢ ( 𝐴 ∩ 𝐶 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐶 }
3		dfin5	⊢ ( 𝐵 ∩ 𝐶 ) = { 𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝐶 }
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) )