Theorem sseq12d 3273

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 3271	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3272	. 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 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:  3sstr3d  3286  3sstr4d  3287  ssdifeq0  3596  relcnvtr  5287  suppfnss  6470  rdgisucinc  6629  oawordriexmid  6716  nnaword  6757  nnawordi  6761  sbthlem2  7241  isbth  7250  nninff  7426  nninfninc  7427  infnninf  7428  infnninfOLD  7429  nnnninf  7430  nnnninfeq  7432  nnnninfeq2  7433  nninfwlpoimlemg  7479  swrdval  11365  ennnfonelemkh  13247  ennnfonelemrnh  13251  isstruct2im  13306  isstruct2r  13307  basis1  15038  baspartn  15041  eltg  15043  metss  15485  isausgren  16288  issubgr  16378  subgrprop3  16383  wkslem1  16441  wkslem2  16442  iswlk  16444  wlkres  16500  eupthseg  16573  0nninf  16908  nnsf  16909  peano4nninf  16910  nninfalllem1  16912  nninfself  16917