Theorem sseq12d 3223

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 3221	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3222	. 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 1372   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  3sstr3d  3236  3sstr4d  3237  ssdifeq0  3542  relcnvtr  5199  rdgisucinc  6461  oawordriexmid  6546  nnaword  6587  nnawordi  6591  sbthlem2  7042  isbth  7051  nninff  7206  nninfninc  7207  infnninf  7208  infnninfOLD  7209  nnnninf  7210  nnnninfeq  7212  nnnninfeq2  7213  nninfwlpoimlemg  7259  ennnfonelemkh  12702  ennnfonelemrnh  12706  isstruct2im  12761  isstruct2r  12762  basis1  14437  baspartn  14440  eltg  14442  metss  14884  0nninf  15805  nnsf  15806  peano4nninf  15807  nninfalllem1  15809  nninfself  15814