Theorem sseq12d 3201

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 3199	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3200	. 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 1364   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  3sstr3d  3214  3sstr4d  3215  ssdifeq0  3520  relcnvtr  5166  rdgisucinc  6410  oawordriexmid  6495  nnaword  6536  nnawordi  6540  sbthlem2  6987  isbth  6996  nninff  7151  infnninf  7152  infnninfOLD  7153  nnnninf  7154  nnnninfeq  7156  nnnninfeq2  7157  nninfwlpoimlemg  7203  ennnfonelemkh  12463  ennnfonelemrnh  12467  isstruct2im  12522  isstruct2r  12523  basis1  14007  baspartn  14010  eltg  14012  metss  14454  0nninf  15215  nnsf  15216  peano4nninf  15217  nninfalllem1  15219  nninfself  15224