Theorem eqtr3 2227

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 2209	. 2 |- ( B = C <-> C = B )
2		eqtr 2225	. 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 1373

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200

This theorem is referenced by:  eueq  2951  euind  2967  reuind  2985  ssprsseq  3806  preqsn  3829  eusv1  4517  funopg  5324  funinsn  5342  foco  5531  funopdmsn  5787  mpofun  6070  enq0tr  7582  lteupri  7765  elrealeu  7977  rereceu  8037  receuap  8777  xrltso  9953  xrlttri3  9954  iseqf1olemab  10684  fsumparts  11896  odd2np1  12299  grpinveu  13485  exmidsbthrlem  16163