Theorem eqtr2 2106

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 2090	. 2 |- ( A = B <-> B = A )
2		eqtr 2105	. 2 |- ( ( B = A /\ A = C ) -> B = C )
3	1, 2	sylanb 278	1 |- ( ( A = B /\ A = C ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081

This theorem is referenced by:  eqvinc  2740  eqvincg  2741  moop2  4078  reusv3i  4281  relop  4586  f0rn0  5205  fliftfun  5575  th3qlem1  6394  enq0ref  6992  enq0tr  6993  genpdisj  7082  addlsub  7848  fsum2dlemstep  10828  0dvds  11094  cncongr1  11363