Theorem eqtr2 2058

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 2042	. 2 |- ( A = B <-> B = A )
2		eqtr 2057	. 2 |- ( ( B = A /\ A = C ) -> B = C )
3	1, 2	sylanb 268	1 |- ( ( A = B /\ A = C ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033

This theorem is referenced by:  eqvinc  2667  eqvincg  2668  moop2  3988  reusv3i  4191  relop  4486  fliftfun  5436  th3qlem1  6208  enq0ref  6531  enq0tr  6532  genpdisj  6621