Metamath Proof Explorer
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24Oct1999)


Ref 
Expression 

Hypotheses 
breqtrd.1 
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 ) 


breqtrd.2 
⊢ ( 𝜑 → 𝐵 = 𝐶 ) 

Assertion 
breqtrd 
⊢ ( 𝜑 → 𝐴 𝑅 𝐶 ) 
Proof
Step 
Hyp 
Ref 
Expression 
1 

breqtrd.1 
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 ) 
2 

breqtrd.2 
⊢ ( 𝜑 → 𝐵 = 𝐶 ) 
3 
2

breq2d 
⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑅 𝐶 ) ) 
4 
1 3

mpbid 
⊢ ( 𝜑 → 𝐴 𝑅 𝐶 ) 