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Metamath Proof Explorer

Theorem coeq12d

Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012)

Ref Expression
Hypotheses coeq12d.1 ⊢ φ → A = B
coeq12d.2 ⊢ φ → C = D
Assertion coeq12d ⊢ φ → A ∘ C = B ∘ D

Proof

Step Hyp Ref Expression
1 coeq12d.1 ⊢ φ → A = B
2 coeq12d.2 ⊢ φ → C = D
3 1 coeq1d ⊢ φ → A ∘ C = B ∘ C
4 2 coeq2d ⊢ φ → B ∘ C = B ∘ D
5 3 4 eqtrd ⊢ φ → A ∘ C = B ∘ D