Theorem eqtr2 2099

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 2083	. 2 |- ( A = B <-> B = A )
2		eqtr 2098	. 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 1284

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 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  eqvinc  2718  eqvincg  2719  moop2  4006  reusv3i  4209  relop  4504  fliftfun  5456  th3qlem1  6231  enq0ref  6623  enq0tr  6624  genpdisj  6713  addlsub  7474  0dvds  10215  cncongr1  10485