Theorem sseq12d 3133

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 3131	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3132	. 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 1332   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  3sstr3d  3146  3sstr4d  3147  ssdifeq0  3450  relcnvtr  5066  rdgisucinc  6290  oawordriexmid  6374  nnaword  6415  nnawordi  6419  sbthlem2  6854  isbth  6863  infnninf  7030  nnnninf  7031  ennnfonelemkh  11961  ennnfonelemrnh  11965  isstruct2im  12008  isstruct2r  12009  ressid2  12057  ressval2  12058  basis1  12253  baspartn  12256  eltg  12260  metss  12702  0nninf  13372  nninff  13373  nnsf  13374  peano4nninf  13375  nninfalllemn  13377  nninfalllem1  13378  nninfself  13384