Theorem eqtr 2098

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 2087	. 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 1284

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 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  eqtr2  2099  eqtr3  2100  sylan9eq  2133  eqvinc  2718  eqvincg  2719  uneqdifeqim  3328  preqsn  3567  dtruex  4302  relresfld  4867  relcoi1  4869  eqer  6161  xpiderm  6200  addlsub  7474  bj-findis  10774