Theorem cnm2m1cnm3

Description: Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018)

	Ref	Expression
Assertion	cnm2m1cnm3	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 − 2 ) − 1 ) = ( 𝐴 − 3 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ )
2		2cnd	⊢ ( 𝐴 ∈ ℂ → 2 ∈ ℂ )
3		1cnd	⊢ ( 𝐴 ∈ ℂ → 1 ∈ ℂ )
4	1 2 3	subsub4d	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 − 2 ) − 1 ) = ( 𝐴 − ( 2 + 1 ) ) )
5		2p1e3	⊢ ( 2 + 1 ) = 3
6	5	a1i	⊢ ( 𝐴 ∈ ℂ → ( 2 + 1 ) = 3 )
7	6	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 − ( 2 + 1 ) ) = ( 𝐴 − 3 ) )
8	4 7	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 − 2 ) − 1 ) = ( 𝐴 − 3 ) )