Theorem eqtr2 2176

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 2159	. 2 |- ( A = B <-> B = A )
2		eqtr 2175	. 2 |- ( ( B = A /\ A = C ) -> B = C )
3	1, 2	sylanb 282	1 |- ( ( A = B /\ A = C ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150

This theorem is referenced by:  eqvinc  2835  eqvincg  2836  moop2  4212  reusv3i  4420  relop  4737  f0rn0  5365  fliftfun  5747  th3qlem1  6583  enq0ref  7354  enq0tr  7355  genpdisj  7444  addlsub  8246  fsum2dlemstep  11335  0dvds  11711  cncongr1  11984