Theorem eqtr 2214

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 2203	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimpar 297	1 |- ( ( A = B /\ B = C ) -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  eqtr2  2215  eqtr3  2216  sylan9eq  2249  eqvinc  2887  eqvincg  2888  uneqdifeqim  3537  preqsn  3806  dtruex  4596  relresfld  5200  relcoi1  5202  eqer  6633  xpider  6674  addlsub  8413  bj-findis  15709