Theorem sylan9ssr 3959

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

Hypotheses

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

Assertion

Ref	Expression
sylan9ssr	⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)

Proof of Theorem sylan9ssr

Step	Hyp	Ref	Expression
1		sylan9ssr.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sylan9ssr.2	. . 3 ⊢ (𝜓 → 𝐵 ⊆ 𝐶)
3	1, 2	sylan9ss 3958	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)
4	3	ancoms 463	1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ss 3930

This theorem is referenced by:  intssuni2  4942  marypha1  9393  cardinfima  10080  cfflb  10242  ssfin4  10293  acsfn  17714  mrelatlub  18617  efgval  19786  islbs3  21256  kgentopon  23663  txlly  23761  sigaclci  34466  bnj1014  35293  topjoin  36764  filnetlem3  36779  poimirlem16  38174  mblfinlem3  38197  sspwimpALT  45524  sspwimpALT2  45527  setrecsres  50364