Theorem sylan9ssr 3936

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 3935	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3906

This theorem is referenced by:  intssuni2  4915  marypha1  9347  cardinfima  10019  cfflb  10181  ssfin4  10232  acsfn  17625  mrelatlub  18528  efgval  19692  islbs3  21153  kgentopon  23503  txlly  23601  sigaclci  34276  bnj1014  35103  topjoin  36547  filnetlem3  36562  poimirlem16  37957  mblfinlem3  37980  sspwimpALT  45351  sspwimpALT2  45354  setrecsres  50177