Theorem eqtr 2099

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 2088	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimpar 291	1 |- ( ( A = B /\ B = C ) -> A = C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-cleq 2075

This theorem is referenced by:  eqtr2  2100  eqtr3  2101  sylan9eq  2134  eqvinc  2719  eqvincg  2720  uneqdifeqim  3335  preqsn  3575  dtruex  4310  relresfld  4877  relcoi1  4879  eqer  6204  xpiderm  6243  addlsub  7541  bj-findis  10932