Theorem sseq2 3048

Description: Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)

Assertion

Ref	Expression
sseq2	|- ( A = B -> ( C C_ A <-> C C_ B ) )

Proof of Theorem sseq2

Step	Hyp	Ref	Expression
1		sstr2 3032	. . . 4 |- ( C C_ A -> ( A C_ B -> C C_ B ) )
2	1	com12 30	. . 3 |- ( A C_ B -> ( C C_ A -> C C_ B ) )
3		sstr2 3032	. . . 4 |- ( C C_ B -> ( B C_ A -> C C_ A ) )
4	3	com12 30	. . 3 |- ( B C_ A -> ( C C_ B -> C C_ A ) )
5	2, 4	anim12i 331	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( ( C C_ A -> C C_ B ) /\ ( C C_ B -> C C_ A ) ) )
6		eqss 3040	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
7		dfbi2 380	. 2 |- ( ( C C_ A <-> C C_ B ) <-> ( ( C C_ A -> C C_ B ) /\ ( C C_ B -> C C_ A ) ) )
8	5, 6, 7	3imtr4i 199	1 |- ( A = B -> ( C C_ A <-> C C_ B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   C_ wss 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  sseq12  3049  sseq2i  3051  sseq2d  3054  syl5sseq  3074  nssne1  3082  sseq0  3324  un00  3329  pweq  3432  ssintab  3705  ssintub  3706  intmin  3708  treq  3942  ssexg  3978  exmidundif  4035  frforeq3  4174  frirrg  4177  iunpw  4302  ordtri2orexmid  4339  ontr2exmid  4341  onsucsssucexmid  4343  ordtri2or2exmid  4387  fununi  5082  funcnvuni  5083  feq3  5147  ssimaexg  5366  nnawordex  6287  ereq1  6299  xpiderm  6363  domeng  6469  ssfiexmid  6592  fisseneq  6642  sbthlemi4  6669  sbthlemi5  6670  bdssexg  11795  bj-nntrans  11846  bj-omtrans  11851