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Metamath Proof Explorer

Theorem sstrd

Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004)

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

Proof

Step Hyp Ref Expression
1 sstrd.1 ⊢ φ → A ⊆ B
2 sstrd.2 ⊢ φ → B ⊆ C
3 sstr ⊢ A ⊆ B ∧ B ⊆ C → A ⊆ C
4 1 2 3 syl2anc ⊢ φ → A ⊆ C