Theorem eqtr2 2248

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
eqtr2	|- ( ( A = B /\ A = C ) -> B = C )

Proof of Theorem eqtr2

Step	Hyp	Ref	Expression
1		eqcom 2231	. 2 |- ( A = B <-> B = A )
2		eqtr 2247	. 2 |- ( ( B = A /\ A = C ) -> B = C )
3	1, 2	sylanb 284	1 |- ( ( A = B /\ A = C ) -> B = C )

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:  eqvinc  2926  eqvincg  2927  moop2  4338  reusv3i  4550  relop  4872  f0rn0  5520  fliftfun  5920  th3qlem1  6784  enq0ref  7620  enq0tr  7621  genpdisj  7710  addlsub  8516  wrd2ind  11255  fsum2dlemstep  11945  0dvds  12322  cncongr1  12625  4sqlem12  12925  uhgr2edg  16004  usgredgreu  16014  uspgredg2vtxeu  16016