Theorem 3sstr4d 3293

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 3279	. 2 |- ( ph -> ( C C_ D <-> A C_ B ) )
5	1, 4	mpbird 167	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  ressuppss  6484  suppfnss  6487  suppssfvg  6493  rdgss  6644  sucinc2  6709  oawordi  6732  nnnninf  7456  fzoss1  10558  fzoss2  10559  swrd0g  11410  lspss  14708  clsss  15142  ntrss  15143  sslm  15271  txss12  15290  metss2lem  15521  xmettxlem  15533  xmettx  15534  plyss  15762  ifpsnprss  16498  nnsf  16953  nninfself  16961