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Metamath Proof Explorer

Theorem subsub

Description: Law for double subtraction. (Contributed by NM, 13-May-2004)

Ref Expression
Assertion subsub ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )

Proof

Step Hyp Ref Expression
1 subsub2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
2 addsubass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
3 addsub ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
4 2 3 eqtr3d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
5 4 3com23 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
6 1 5 eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )