Theorem sseq12 3022

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 3020	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2		sseq2 3021	. 2 |- ( C = D -> ( B C_ C <-> B C_ D ) )
3	1, 2	sylan9bb 449	1 |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )

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:  sseq12i  3025  funcnvuni  4988  fun11iun  5167