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Metamath Proof Explorer

Theorem 3eqtrrd

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006) (Proof shortened by Andrew Salmon, 25-May-2011)

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

Proof

Step Hyp Ref Expression
1 3eqtrd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 3eqtrd.2 ⊢ ( 𝜑 → 𝐵 = 𝐶 )
3 3eqtrd.3 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
4 1 2 eqtrd ⊢ ( 𝜑 → 𝐴 = 𝐶 )
5 4 3 eqtr2d ⊢ ( 𝜑 → 𝐷 = 𝐴 )