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Metamath Proof Explorer

Theorem sylan9ssr

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004)

Ref Expression
Hypotheses sylan9ssr.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
sylan9ssr.2 ⊢ ( 𝜓 → 𝐵 ⊆ 𝐶 )
Assertion sylan9ssr ( ( 𝜓 ∧ 𝜑 ) → 𝐴 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 sylan9ssr.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2 sylan9ssr.2 ⊢ ( 𝜓 → 𝐵 ⊆ 𝐶 )
3 1 2 sylan9ss ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ⊆ 𝐶 )
4 3 ancoms ⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝐴 ⊆ 𝐶 )