Theorem eqtr 2256

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 2245	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2	1	biimpar 297	1 |- ( ( A = B /\ B = C ) -> A = C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqtr2  2257  eqtr3  2258  sylan9eq  2291  eqvinc  2949  eqvincg  2950  uneqdifeqim  3613  preqsn  3900  dtruex  4706  relresfld  5317  relcoi1  5319  eqer  6839  xpider  6880  addlsub  8696  uhgr2edg  16447  bj-findis  17005