Theorem sseq12 3204

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 3202	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2		sseq2 3203	. 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 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:  sseq12i  3207  undifexmid  4222  exmidundif  4235  exmidundifim  4236  funcnvuni  5323  fun11iun  5521