Theorem sseq2 3021

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 3006	. . . 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 3006	. . . 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 3014	. 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 1284   C_ wss 2973

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  sseq12  3022  sseq2i  3024  sseq2d  3027  syl5sseq  3047  nssne1  3055  sseq0  3285  un00  3290  pweq  3385  ssintab  3653  ssintub  3654  intmin  3656  treq  3881  ssexg  3917  frforeq3  4102  frirrg  4105  iunpw  4229  ordtri2orexmid  4266  ontr2exmid  4268  onsucsssucexmid  4270  ordtri2or2exmid  4314  fununi  4987  funcnvuni  4988  feq3  5052  ssimaexg  5256  nnawordex  6124  ereq1  6136  xpiderm  6200  domeng  6256  ssfiexmid  6361  bdssexg  10695  bj-nntrans  10746  bj-omtrans  10751