Theorem sseq12 3172

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 3170	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2		sseq2 3171	. 2 |- ( C = D -> ( B C_ C <-> B C_ D ) )
3	1, 2	sylan9bb 459	1 |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  sseq12i  3175  undifexmid  4179  exmidundif  4192  exmidundifim  4193  funcnvuni  5267  fun11iun  5463