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 ) → ( ( 𝐴 − 𝐵 ) − 𝐶 ) = ( ( 𝐴 − 𝐶 ) − 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 ) → ( ( 𝐴 − 𝐵 ) − 𝐶 ) = ( 𝐴 − ( 𝐵 + 𝐶 ) ) )
5		subsub4	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐶 ) − 𝐵 ) = ( 𝐴 − ( 𝐶 + 𝐵 ) ) )
6	5	3com23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐶 ) − 𝐵 ) = ( 𝐴 − ( 𝐶 + 𝐵 ) ) )
7	3 4 6	3eqtr4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) − 𝐶 ) = ( ( 𝐴 − 𝐶 ) − 𝐵 ) )