Metamath Proof Explorer


Theorem 3eqtr3g

Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994)

Ref Expression
Hypotheses 3eqtr3g.1 ( 𝜑𝐴 = 𝐵 )
3eqtr3g.2 𝐴 = 𝐶
3eqtr3g.3 𝐵 = 𝐷
Assertion 3eqtr3g ( 𝜑𝐶 = 𝐷 )

Proof

Step Hyp Ref Expression
1 3eqtr3g.1 ( 𝜑𝐴 = 𝐵 )
2 3eqtr3g.2 𝐴 = 𝐶
3 3eqtr3g.3 𝐵 = 𝐷
4 2 1 eqtr3id ( 𝜑𝐶 = 𝐵 )
5 4 3 eqtrdi ( 𝜑𝐶 = 𝐷 )