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Metamath Proof Explorer

Theorem 3eqtr4g

Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypotheses 3eqtr4g.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
3eqtr4g.2 ⊢ 𝐶 = 𝐴
3eqtr4g.3 ⊢ 𝐷 = 𝐵
Assertion 3eqtr4g ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step Hyp Ref Expression
1 3eqtr4g.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 3eqtr4g.2 ⊢ 𝐶 = 𝐴
3 3eqtr4g.3 ⊢ 𝐷 = 𝐵
4 2 1 eqtrid ⊢ ( 𝜑 → 𝐶 = 𝐵 )
5 4 3 eqtr4di ⊢ ( 𝜑 → 𝐶 = 𝐷 )