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Metamath Proof Explorer

Theorem sseldd

Description: Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004)

Ref Expression
Hypotheses sseld.1 ⊢ φ → A ⊆ B
sseldd.2 ⊢ φ → C ∈ A
Assertion sseldd ⊢ φ → C ∈ B

Proof

Step Hyp Ref Expression
1 sseld.1 ⊢ φ → A ⊆ B
2 sseldd.2 ⊢ φ → C ∈ A
3 1 sseld ⊢ φ → C ∈ A → C ∈ B
4 2 3 mpd ⊢ φ → C ∈ B