Theorem sseq2 3293

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 3279	. . . 4 |- (C C_ A -> (A C_ B -> C C_ B))
2	1	com12 27	. . 3 |- (A C_ B -> (C C_ A -> C C_ B))
3		sstr2 3279	. . . 4 |- (C C_ B -> (B C_ A -> C C_ A))
4	3	com12 27	. . 3 |- (B C_ A -> (C C_ B -> C C_ A))
5	2, 4	anim12i 549	. 2 |- ((A C_ B /\ B C_ A) -> ((C C_ A -> C C_ B) /\ (C C_ B -> C C_ A)))
6		eqss 3287	. 2 |- (A = B <-> (A C_ B /\ B C_ A))
7		dfbi2 609	. 2 |- ((C C_ A <-> C C_ B) <-> ((C C_ A -> C C_ B) /\ (C C_ B -> C C_ A)))
8	5, 6, 7	3imtr4i 257	1 |- (A = B -> (C C_ A <-> C C_ B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  sseq12  3294  sseq2i  3296  sseq2d  3299  syl5sseq  3319  nssne1  3327  psseq2  3357  sseq0  3582  un00  3586  disjpss  3601  pweq  3725  ssintab  3943  ssintub  3944  intmin  3946  p6eq  4238  opkelssetkg  4268  pw1equn  4331  pw1eqadj  4332  ssfin  4470  sfinltfin  4535  vfinspsslem1  4550  brssetg  4757  fununi  5160  funcnvuni  5161  feq3  5212  clos1induct  5880  frd  5922  nclec  6195  lecidg  6196  lecncvg  6199  ltcpw1pwg  6202  sbthlem2  6204  addlec  6208  nc0le1  6216  ce2le  6233  tlenc1c  6240