MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sylan9ssr Structured version   Visualization version   GIF version

Theorem sylan9ssr 3963
Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
Hypotheses
Ref Expression
sylan9ssr.1 (𝜑𝐴𝐵)
sylan9ssr.2 (𝜓𝐵𝐶)
Assertion
Ref Expression
sylan9ssr ((𝜓𝜑) → 𝐴𝐶)

Proof of Theorem sylan9ssr
StepHypRef Expression
1 sylan9ssr.1 . . 3 (𝜑𝐴𝐵)
2 sylan9ssr.2 . . 3 (𝜓𝐵𝐶)
31, 2sylan9ss 3962 . 2 ((𝜑𝜓) → 𝐴𝐶)
43ancoms 460 1 ((𝜓𝜑) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wss 3915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932
This theorem is referenced by:  intssuni2  4939  marypha1  9377  cardinfima  10040  cfflb  10202  ssfin4  10253  acsfn  17546  mrelatlub  18458  efgval  19506  islbs3  20632  kgentopon  22905  txlly  23003  sigaclci  32771  bnj1014  33613  topjoin  34866  filnetlem3  34881  poimirlem16  36123  mblfinlem3  36146  sspwimpALT  43281  sspwimpALT2  43284  setrecsres  47221
  Copyright terms: Public domain W3C validator