Metamath Proof Explorer
Description: A deduction from three chained equalities. (Contributed by NM, 4Aug2006)


Ref 
Expression 

Hypotheses 
3eqtr2d.1 
⊢ ( 𝜑 → 𝐴 = 𝐵 ) 


3eqtr2d.2 
⊢ ( 𝜑 → 𝐶 = 𝐵 ) 


3eqtr2d.3 
⊢ ( 𝜑 → 𝐶 = 𝐷 ) 

Assertion 
3eqtr2d 
⊢ ( 𝜑 → 𝐴 = 𝐷 ) 
Proof
Step 
Hyp 
Ref 
Expression 
1 

3eqtr2d.1 
⊢ ( 𝜑 → 𝐴 = 𝐵 ) 
2 

3eqtr2d.2 
⊢ ( 𝜑 → 𝐶 = 𝐵 ) 
3 

3eqtr2d.3 
⊢ ( 𝜑 → 𝐶 = 𝐷 ) 
4 
1 2

eqtr4d 
⊢ ( 𝜑 → 𝐴 = 𝐶 ) 
5 
4 3

eqtrd 
⊢ ( 𝜑 → 𝐴 = 𝐷 ) 