Theorem eqtr2 2158

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 2141	. 2 |- ( A = B <-> B = A )
2		eqtr 2157	. 2 |- ( ( B = A /\ A = C ) -> B = C )
3	1, 2	sylanb 282	1 |- ( ( A = B /\ A = C ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by:  eqvinc  2808  eqvincg  2809  moop2  4173  reusv3i  4380  relop  4689  f0rn0  5317  fliftfun  5697  th3qlem1  6531  enq0ref  7241  enq0tr  7242  genpdisj  7331  addlsub  8132  fsum2dlemstep  11203  0dvds  11513  cncongr1  11784