Theorem eqtr 2155

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 2144	. 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 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 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130

This theorem is referenced by:  eqtr2  2156  eqtr3  2157  sylan9eq  2190  eqvinc  2803  eqvincg  2804  uneqdifeqim  3443  preqsn  3697  dtruex  4469  relresfld  5063  relcoi1  5065  eqer  6454  xpider  6493  addlsub  8125  bj-findis  13166