Theorem 3sstr4d 3215

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 3201	. 2 |- ( ph -> ( C C_ D <-> A C_ B ) )
5	1, 4	mpbird 167	1 |- ( ph -> C C_ D )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  rdgss  6407  sucinc2  6470  oawordi  6493  nnnninf  7153  fzoss1  10200  fzoss2  10201  lspss  13712  clsss  14070  ntrss  14071  sslm  14199  txss12  14218  metss2lem  14449  xmettxlem  14461  xmettx  14462  nnsf  15208  nninfself  15216