Theorem eqtr 2057

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 2046	. 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 1243

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 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033

This theorem is referenced by:  eqtr2  2058  eqtr3  2059  sylan9eq  2092  eqvinc  2667  eqvincg  2668  uneqdifeqim  3308  preqsn  3546  dtruex  4283  relresfld  4847  relcoi1  4849  eqer  6138  xpiderm  6177  bj-findis  10104