Theorem sylan9ssr 3973

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 3972	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3943

This theorem is referenced by:  intssuni2  4949  marypha1  9446  cardinfima  10111  cfflb  10273  ssfin4  10324  acsfn  17671  mrelatlub  18572  efgval  19698  islbs3  21116  kgentopon  23476  txlly  23574  sigaclci  34163  bnj1014  34992  topjoin  36383  filnetlem3  36398  poimirlem16  37660  mblfinlem3  37683  sspwimpALT  44949  sspwimpALT2  44952  setrecsres  49566