Theorem sseq12d 3128

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 3126	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3127	. 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 1331   C_ wss 3071

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  3sstr3d  3141  3sstr4d  3142  ssdifeq0  3445  relcnvtr  5058  rdgisucinc  6282  oawordriexmid  6366  nnaword  6407  nnawordi  6411  sbthlem2  6846  isbth  6855  infnninf  7022  nnnninf  7023  ennnfonelemkh  11925  ennnfonelemrnh  11929  isstruct2im  11969  isstruct2r  11970  ressid2  12018  ressval2  12019  basis1  12214  baspartn  12217  eltg  12221  metss  12663  0nninf  13197  nninff  13198  nnsf  13199  peano4nninf  13200  nninfalllemn  13202  nninfalllem1  13203  nninfself  13209