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  5248  rdgisucinc  6537  oawordriexmid  6624  nnaword  6665  nnawordi  6669  sbthlem2  7136  isbth  7145  nninff  7300  nninfninc  7301  infnninf  7302  infnninfOLD  7303  nnnninf  7304  nnnninfeq  7306  nnnninfeq2  7307  nninfwlpoimlemg  7353  swrdval  11195  ennnfonelemkh  12998  ennnfonelemrnh  13002  isstruct2im  13057  isstruct2r  13058  basis1  14736  baspartn  14739  eltg  14741  metss  15183  isausgren  15980  wkslem1  16061  wkslem2  16062  iswlk  16064  wlkres  16118  0nninf  16430  nnsf  16431  peano4nninf  16432  nninfalllem1  16434  nninfself  16439