Theorem eqtr2 2215

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 2198	. 2 |- ( A = B <-> B = A )
2		eqtr 2214	. 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 1364

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  eqvinc  2887  eqvincg  2888  moop2  4284  reusv3i  4494  relop  4816  f0rn0  5452  fliftfun  5843  th3qlem1  6696  enq0ref  7500  enq0tr  7501  genpdisj  7590  addlsub  8396  fsum2dlemstep  11599  0dvds  11976  cncongr1  12271  4sqlem12  12571