Theorem eqtr2 2196

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 2179	. 2 |- ( A = B <-> B = A )
2		eqtr 2195	. 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 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  eqvinc  2860  eqvincg  2861  moop2  4251  reusv3i  4459  relop  4777  f0rn0  5410  fliftfun  5796  th3qlem1  6636  enq0ref  7431  enq0tr  7432  genpdisj  7521  addlsub  8325  fsum2dlemstep  11437  0dvds  11813  cncongr1  12097