Theorem 3sstr4d 3287

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4d.1	|- ( ph -> A C_ B )
3sstr4d.2	|- ( ph -> C = A )
3sstr4d.3	|- ( ph -> D = B )

Assertion

Ref	Expression
3sstr4d	|- ( ph -> C C_ D )

Proof of Theorem 3sstr4d

Step	Hyp	Ref	Expression
1		3sstr4d.1	. 2 |- ( ph -> A C_ B )
2		3sstr4d.2	. . 3 |- ( ph -> C = A )
3		3sstr4d.3	. . 3 |- ( ph -> D = B )
4	2, 3	sseq12d 3273	. 2 |- ( ph -> ( C C_ D <-> A C_ B ) )
5	1, 4	mpbird 167	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  ressuppss  6468  suppfnss  6471  suppssfvg  6477  rdgss  6628  sucinc2  6693  oawordi  6716  nnnninf  7431  fzoss1  10533  fzoss2  10534  swrd0g  11381  lspss  14678  clsss  15114  ntrss  15115  sslm  15243  txss12  15262  metss2lem  15493  xmettxlem  15505  xmettx  15506  plyss  15734  ifpsnprss  16469  nnsf  16924  nninfself  16932