Metamath Proof Explorer


Theorem inabs3

Description: Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion inabs3 ( 𝐶𝐵 → ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ( 𝐴𝐶 ) )

Proof

Step Hyp Ref Expression
1 inass ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ( 𝐴 ∩ ( 𝐵𝐶 ) )
2 sseqin2 ( 𝐶𝐵 ↔ ( 𝐵𝐶 ) = 𝐶 )
3 2 biimpi ( 𝐶𝐵 → ( 𝐵𝐶 ) = 𝐶 )
4 3 ineq2d ( 𝐶𝐵 → ( 𝐴 ∩ ( 𝐵𝐶 ) ) = ( 𝐴𝐶 ) )
5 1 4 syl5eq ( 𝐶𝐵 → ( ( 𝐴𝐵 ) ∩ 𝐶 ) = ( 𝐴𝐶 ) )