Theorem eqtr 2183

Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)

Assertion

Ref	Expression
eqtr	|- ( ( A = B /\ B = C ) -> A = C )

Proof of Theorem eqtr

Step	Hyp	Ref	Expression
1		eqeq1 2172	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimpar 295	1 |- ( ( A = B /\ B = C ) -> A = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  eqtr2  2184  eqtr3  2185  sylan9eq  2219  eqvinc  2849  eqvincg  2850  uneqdifeqim  3494  preqsn  3755  dtruex  4536  relresfld  5133  relcoi1  5135  eqer  6533  xpider  6572  addlsub  8268  bj-findis  13861