Theorem sseq12 3628

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 3626	. 2 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2		sseq2 3627	. 2 |- ( C = D -> ( B C_ C <-> B C_ D ) )
3	1, 2	sylan9bb 736	1 |- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by:  sseq12i  3631  sorpsscmpl  6948  funcnvuni  7119  fun11iun  7126  sornom  9099  axdc3lem2  9273  ipole  17158  ipodrsima  17165  cmetss  23113  funpsstri  31663  ismrcd2  37262  ismrc  37264