Theorem pncan

Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
2		simpl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ )
3	1 2	addcomd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 + 𝐴 ) = ( 𝐴 + 𝐵 ) )
4		addcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
5		subadd	⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( ( 𝐴 + 𝐵 ) − 𝐵 ) = 𝐴 ↔ ( 𝐵 + 𝐴 ) = ( 𝐴 + 𝐵 ) ) )
6	4 1 2 5	syl3anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐴 + 𝐵 ) − 𝐵 ) = 𝐴 ↔ ( 𝐵 + 𝐴 ) = ( 𝐴 + 𝐵 ) ) )
7	3 6	mpbird	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐵 ) = 𝐴 )