Theorem eqtr 2223

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 2212	. 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 1373

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198

This theorem is referenced by:  eqtr2  2224  eqtr3  2225  sylan9eq  2258  eqvinc  2896  eqvincg  2897  uneqdifeqim  3546  preqsn  3816  dtruex  4608  relresfld  5213  relcoi1  5215  eqer  6654  xpider  6695  addlsub  8444  bj-findis  15952