Theorem sseq12d 3210

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 3208	. 2 |- ( ph -> ( A C_ C <-> B C_ C ) )
3		sseq12d.2	. . 3 |- ( ph -> C = D )
4	3	sseq2d 3209	. 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 3153

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 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  3sstr3d  3223  3sstr4d  3224  ssdifeq0  3529  relcnvtr  5185  rdgisucinc  6438  oawordriexmid  6523  nnaword  6564  nnawordi  6568  sbthlem2  7017  isbth  7026  nninff  7181  nninfninc  7182  infnninf  7183  infnninfOLD  7184  nnnninf  7185  nnnninfeq  7187  nnnninfeq2  7188  nninfwlpoimlemg  7234  ennnfonelemkh  12569  ennnfonelemrnh  12573  isstruct2im  12628  isstruct2r  12629  basis1  14215  baspartn  14218  eltg  14220  metss  14662  0nninf  15494  nnsf  15495  peano4nninf  15496  nninfalllem1  15498  nninfself  15503