Theorem eqtr 2054

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 2043	. 2 |- (A = B -> (A = C <-> B = C))
2	1	biimpar 281	1 |- ((A = B /\ B = C) -> A = C)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030

This theorem is referenced by:  eqtr2  2055  eqtr3  2056  sylan9eq  2089  eqvinc  2661  eqvincg  2662  uneqdifeqim  3302  preqsn  3537  dtruex  4237  relresfld  4790  relcoi1  4792  eqer  6074  xpiderm  6113  bj-findis  9439