Theorem 3eqtr3rd 2124

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 2117	. 2 |- ( ph -> B = C )
5	1, 4	eqtr3d 2117	1 |- ( ph -> D = C )

Syntax hints:   -> wi 4   = wceq 1285

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-cleq 2076

This theorem is referenced by:  fcofo  5503  fcof1o  5508  frecabcl  6096  nnaword  6200  enomnilem  6699  fodjuomnilem0  6707  pn0sr  7220  negeu  7576  add20  7855  2halves  8537  bcnn  10000  bcpasc  10009  resqrexlemover  10270  gcdid  10757  phiprmpw  10978