Metamath Proof Explorer


Theorem sub32

Description: Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005)

Ref Expression
Assertion sub32 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴𝐵 ) − 𝐶 ) = ( ( 𝐴𝐶 ) − 𝐵 ) )

Proof

Step Hyp Ref Expression
1 addcom ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
2 1 3adant1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
3 2 oveq2d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 + 𝐶 ) ) = ( 𝐴 − ( 𝐶 + 𝐵 ) ) )
4 subsub4 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴𝐵 ) − 𝐶 ) = ( 𝐴 − ( 𝐵 + 𝐶 ) ) )
5 subsub4 ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴𝐶 ) − 𝐵 ) = ( 𝐴 − ( 𝐶 + 𝐵 ) ) )
6 5 3com23 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴𝐶 ) − 𝐵 ) = ( 𝐴 − ( 𝐶 + 𝐵 ) ) )
7 3 4 6 3eqtr4d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴𝐵 ) − 𝐶 ) = ( ( 𝐴𝐶 ) − 𝐵 ) )