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Metamath Proof Explorer

Theorem ineq2

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

Ref Expression
Assertion ineq2 ( 𝐴 = 𝐵 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ineq1 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) )
2 incom ⊢ ( 𝐶 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐶 )
3 incom ⊢ ( 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐶 )
4 1 2 3 3eqtr4g ⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )