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Metamath Proof Explorer

Theorem eleqtrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Proof

Step Hyp Ref Expression
1 eleqtrd.1 ⊢ φ → A ∈ B
2 eleqtrd.2 ⊢ φ → B = C
3 2 eleq2d ⊢ φ → A ∈ B ↔ A ∈ C
4 1 3 mpbid ⊢ φ → A ∈ C