Theorem eqtr 2252

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 2241	. 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 1398

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  eqtr2  2253  eqtr3  2254  sylan9eq  2287  eqvinc  2942  eqvincg  2943  uneqdifeqim  3597  preqsn  3881  dtruex  4683  relresfld  5294  relcoi1  5296  eqer  6801  xpider  6842  addlsub  8645  uhgr2edg  16218  bj-findis  16766