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Metamath Proof Explorer

Theorem eqeltrd

Description: Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002)

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

Proof

Step Hyp Ref Expression
1 eqeltrd.1 ⊢ φ → A = B
2 eqeltrd.2 ⊢ φ → B ∈ C
3 1 eleq1d ⊢ φ → A ∈ C ↔ B ∈ C
4 2 3 mpbird ⊢ φ → A ∈ C