MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssrd Structured version   Unicode version

Theorem ssrd 3494
Description: Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Hypotheses
Ref Expression
ssrd.0  |-  F/ x ph
ssrd.1  |-  F/_ x A
ssrd.2  |-  F/_ x B
ssrd.3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )
Assertion
Ref Expression
ssrd  |-  ( ph  ->  A  C_  B )

Proof of Theorem ssrd
StepHypRef Expression
1 ssrd.0 . . 3  |-  F/ x ph
2 ssrd.3 . . 3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )
31, 2alrimi 1882 . 2  |-  ( ph  ->  A. x ( x  e.  A  ->  x  e.  B ) )
4 ssrd.1 . . 3  |-  F/_ x A
5 ssrd.2 . . 3  |-  F/_ x B
64, 5dfss2f 3480 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
73, 6sylibr 212 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1396   F/wnf 1621    e. wcel 1823   F/_wnfc 2602    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-in 3468  df-ss 3475
This theorem is referenced by:  eqrd  3507  neiptopnei  19800  rabss3d  27611  iunxdif3  27637  ssfiunibd  31748  stoweidlem52  32073  stoweidlem59  32080
  Copyright terms: Public domain W3C validator