Theorem 3eqtr3rd 2081

Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)

Hypotheses

Ref	Expression
3eqtr3d.1	|- ( ph -> A = B )
3eqtr3d.2	|- ( ph -> A = C )
3eqtr3d.3	|- ( ph -> B = D )

Assertion

Ref	Expression
3eqtr3rd	|- ( ph -> D = C )

Proof of Theorem 3eqtr3rd

Step	Hyp	Ref	Expression
1		3eqtr3d.3	. 2 |- ( ph -> B = D )
2		3eqtr3d.1	. . 3 |- ( ph -> A = B )
3		3eqtr3d.2	. . 3 |- ( ph -> A = C )
4	2, 3	eqtr3d 2074	. 2 |- ( ph -> B = C )
5	1, 4	eqtr3d 2074	1 |- ( ph -> D = C )

Syntax hints:   -> wi 4   = 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:  fcofo  5424  fcof1o  5429  nnaword  6084  pn0sr  6856  negeu  7202  add20  7469  2halves  8154  resqrexlemover  9608