Theorem eqtr 2157

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 2146	. 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 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by:  eqtr2  2158  eqtr3  2159  sylan9eq  2192  eqvinc  2808  eqvincg  2809  uneqdifeqim  3448  preqsn  3702  dtruex  4474  relresfld  5068  relcoi1  5070  eqer  6461  xpider  6500  addlsub  8132  bj-findis  13177