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Metamath Proof Explorer

Theorem eqtrd

Description: An equality transitivity deduction. (Contributed by NM, 21-Jun-1993)

Ref Expression
Hypotheses eqtrd.1 ⊢ φ → A = B
eqtrd.2 ⊢ φ → B = C
Assertion eqtrd ⊢ φ → A = C

Proof

Step Hyp Ref Expression
1 eqtrd.1 ⊢ φ → A = B
2 eqtrd.2 ⊢ φ → B = C
3 2 eqeq2d ⊢ φ → A = B ↔ A = C
4 1 3 mpbid ⊢ φ → A = C