Theorem eqtr 2158

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 2147	. 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 1332

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 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133

This theorem is referenced by:  eqtr2  2159  eqtr3  2160  sylan9eq  2193  eqvinc  2812  eqvincg  2813  uneqdifeqim  3453  preqsn  3710  dtruex  4482  relresfld  5076  relcoi1  5078  eqer  6469  xpider  6508  addlsub  8156  bj-findis  13348