Theorem sseq12 3249

Description: Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)

Assertion

Ref	Expression
sseq12	|- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )

Proof of Theorem sseq12

Step	Hyp	Ref	Expression
1		sseq1 3247	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2		sseq2 3248	. 2 |- ( C = D -> ( B C_ C <-> B C_ D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  sseq12i  3252  undifexmid  4276  exmidundif  4289  exmidundifim  4290  funcnvuni  5389  fun11iun  5592