Theorem eqtr3 2249

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.)

Assertion

Ref	Expression
eqtr3	|- ( ( A = C /\ B = C ) -> A = B )

Proof of Theorem eqtr3

Step	Hyp	Ref	Expression
1		eqcom 2231	. 2 |- ( B = C <-> C = B )
2		eqtr 2247	. 2 |- ( ( A = C /\ C = B ) -> A = B )
3	1, 2	sylan2b 287	1 |- ( ( A = C /\ B = C ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395

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-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eueq  2974  euind  2990  reuind  3008  ssprsseq  3830  preqsn  3853  eusv1  4544  funopg  5355  funinsn  5373  foco  5564  funopdmsn  5826  mpofun  6115  enq0tr  7637  lteupri  7820  elrealeu  8032  rereceu  8092  receuap  8832  xrltso  10009  xrlttri3  10010  iseqf1olemab  10741  fsumparts  12002  odd2np1  12405  grpinveu  13592  exmidsbthrlem  16504