Theorem eqtr2 2212

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 2195	. 2 |- ( A = B <-> B = A )
2		eqtr 2211	. 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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  eqvinc  2883  eqvincg  2884  moop2  4280  reusv3i  4490  relop  4812  f0rn0  5448  fliftfun  5839  th3qlem1  6691  enq0ref  7493  enq0tr  7494  genpdisj  7583  addlsub  8389  fsum2dlemstep  11577  0dvds  11954  cncongr1  12241  4sqlem12  12540