Theorem sseq12d 3173

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 3171	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3172	. 2 |- ( ph -> ( B C_ C <-> B C_ D ) )
5	2, 4	bitrd 187	1 |- ( ph -> ( A C_ C <-> B C_ D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  3sstr3d  3186  3sstr4d  3187  ssdifeq0  3491  relcnvtr  5123  rdgisucinc  6353  oawordriexmid  6438  nnaword  6479  nnawordi  6483  sbthlem2  6923  isbth  6932  nninff  7087  infnninf  7088  infnninfOLD  7089  nnnninf  7090  nnnninfeq  7092  nnnninfeq2  7093  ennnfonelemkh  12345  ennnfonelemrnh  12349  isstruct2im  12404  isstruct2r  12405  ressid2  12454  ressval2  12455  basis1  12685  baspartn  12688  eltg  12692  metss  13134  0nninf  13884  nnsf  13885  peano4nninf  13886  nninfalllem1  13888  nninfself  13893