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Metamath Proof Explorer

Theorem eqsstrd

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004)

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

Proof

Step Hyp Ref Expression
1 eqsstrd.1 ⊢ φ → A = B
2 eqsstrd.2 ⊢ φ → B ⊆ C
3 1 sseq1d ⊢ φ → A ⊆ C ↔ B ⊆ C
4 2 3 mpbird ⊢ φ → A ⊆ C