Metamath Proof Explorer
Description: Subtraction and addition of equals. (Contributed by Mario Carneiro, 27May2016)


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 ) 