Theorem eqtr 2188

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 2177	. 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 1348

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 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163

This theorem is referenced by:  eqtr2  2189  eqtr3  2190  sylan9eq  2223  eqvinc  2853  eqvincg  2854  uneqdifeqim  3500  preqsn  3762  dtruex  4543  relresfld  5140  relcoi1  5142  eqer  6545  xpider  6584  addlsub  8289  bj-findis  14014