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Metamath Proof Explorer

Theorem ineq1i

Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993)

Ref Expression
Hypothesis ineq1i.1 ⊢ 𝐴 = 𝐵
Assertion ineq1i ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 )

Proof

Step Hyp Ref Expression
1 ineq1i.1 ⊢ 𝐴 = 𝐵
2 ineq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) )
3 1 2 ax-mp ⊢ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 )