Theorem pncan3d

Description: Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	pncan3d	⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 − 𝐴 ) ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		pncan3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + ( 𝐵 − 𝐴 ) ) = 𝐵 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 − 𝐴 ) ) = 𝐵 )