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Metamath Proof Explorer

Theorem eqsstrid

Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004)

Ref Expression
Hypotheses eqsstrid.1 ⊢ A = B
eqsstrid.2 ⊢ φ → B ⊆ C
Assertion eqsstrid ⊢ φ → A ⊆ C

Proof

Step Hyp Ref Expression
1 eqsstrid.1 ⊢ A = B
2 eqsstrid.2 ⊢ φ → B ⊆ C
3 1 sseq1i ⊢ A ⊆ C ↔ B ⊆ C
4 2 3 sylibr ⊢ φ → A ⊆ C