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Metamath Proof Explorer

Theorem eleqtrrdi

Description: A membership and equality inference. (Contributed by NM, 24-Apr-2005)

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

Proof

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