Theorem eqtr2 2224

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 2207	. 2 |- ( A = B <-> B = A )
2		eqtr 2223	. 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 1373

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198

This theorem is referenced by:  eqvinc  2896  eqvincg  2897  moop2  4296  reusv3i  4506  relop  4828  f0rn0  5470  fliftfun  5865  th3qlem1  6724  enq0ref  7546  enq0tr  7547  genpdisj  7636  addlsub  8442  fsum2dlemstep  11745  0dvds  12122  cncongr1  12425  4sqlem12  12725