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  4339  reusv3i  4551  relop  4875  f0rn0  5525  fliftfun  5929  th3qlem1  6797  enq0ref  7636  enq0tr  7637  genpdisj  7726  addlsub  8532  wrd2ind  11276  fsum2dlemstep  11966  0dvds  12343  cncongr1  12646  4sqlem12  12946  uhgr2edg  16025  usgredgreu  16035  uspgredg2vtxeu  16037