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Metamath Proof Explorer

Theorem eleqtrdi

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006)

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

Proof

Step Hyp Ref Expression
1 eleqtrdi.1 ⊢ φ → A ∈ B
2 eleqtrdi.2 ⊢ B = C
3 2 a1i ⊢ φ → B = C
4 1 3 eleqtrd ⊢ φ → A ∈ C