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Metamath Proof Explorer

Theorem 3eqtr3a

Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015)

Ref Expression
Hypotheses 3eqtr3a.1 ⊢ 𝐴 = 𝐵
3eqtr3a.2 ⊢ ( 𝜑 → 𝐴 = 𝐶 )
3eqtr3a.3 ⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion 3eqtr3a ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step Hyp Ref Expression
1 3eqtr3a.1 ⊢ 𝐴 = 𝐵
2 3eqtr3a.2 ⊢ ( 𝜑 → 𝐴 = 𝐶 )
3 3eqtr3a.3 ⊢ ( 𝜑 → 𝐵 = 𝐷 )
4 1 3 eqtrid ⊢ ( 𝜑 → 𝐴 = 𝐷 )
5 2 4 eqtr3d ⊢ ( 𝜑 → 𝐶 = 𝐷 )