Theorem sseq12d 3255

Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Hypotheses

Ref	Expression
sseq1d.1	|- ( ph -> A = B )
sseq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
sseq12d	|- ( ph -> ( A C_ C <-> B C_ D ) )

Proof of Theorem sseq12d

Step	Hyp	Ref	Expression
1		sseq1d.1	. . 3 |- ( ph -> A = B )
2	1	sseq1d 3253	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3254	. 2 |- ( ph -> ( B C_ C <-> B C_ D ) )
5	2, 4	bitrd 188	1 |- ( ph -> ( A C_ C <-> B C_ D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  3sstr3d  3268  3sstr4d  3269  ssdifeq0  3574  relcnvtr  5247  rdgisucinc  6529  oawordriexmid  6614  nnaword  6655  nnawordi  6659  sbthlem2  7121  isbth  7130  nninff  7285  nninfninc  7286  infnninf  7287  infnninfOLD  7288  nnnninf  7289  nnnninfeq  7291  nnnninfeq2  7292  nninfwlpoimlemg  7338  swrdval  11175  ennnfonelemkh  12978  ennnfonelemrnh  12982  isstruct2im  13037  isstruct2r  13038  basis1  14715  baspartn  14718  eltg  14720  metss  15162  isausgren  15959  wkslem1  16026  wkslem2  16027  iswlk  16029  0nninf  16329  nnsf  16330  peano4nninf  16331  nninfalllem1  16333  nninfself  16338