Theorem sylan9ssr 3961

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 3960	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3914

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 3931

This theorem is referenced by:  intssuni2  4937  marypha1  9385  cardinfima  10050  cfflb  10212  ssfin4  10263  acsfn  17620  mrelatlub  18521  efgval  19647  islbs3  21065  kgentopon  23425  txlly  23523  sigaclci  34122  bnj1014  34951  topjoin  36353  filnetlem3  36368  poimirlem16  37630  mblfinlem3  37653  sspwimpALT  44914  sspwimpALT2  44917  setrecsres  49691