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Metamath Proof Explorer

Theorem eleqtrrid

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

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

Proof

Step Hyp Ref Expression
1 eleqtrrid.1 ⊢ A ∈ B
2 eleqtrrid.2 ⊢ φ → C = B
3 2 eqcomd ⊢ φ → B = C
4 1 3 eleqtrid ⊢ φ → A ∈ C