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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