Theorem 3eqtr2d 2127

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3eqtr2d.1	|- ( ph -> A = B )
3eqtr2d.2	|- ( ph -> C = B )
3eqtr2d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
3eqtr2d	|- ( ph -> A = D )

Proof of Theorem 3eqtr2d

Step	Hyp	Ref	Expression
1		3eqtr2d.1	. . 3 |- ( ph -> A = B )
2		3eqtr2d.2	. . 3 |- ( ph -> C = B )
3	1, 2	eqtr4d 2124	. 2 |- ( ph -> A = C )
4		3eqtr2d.3	. 2 |- ( ph -> C = D )
5	3, 4	eqtrd 2121	1 |- ( ph -> A = D )

Syntax hints:   -> wi 4   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  fmptapd  5502  rdgisucinc  6164  mulidnq  7002  ltrnqg  7033  recexprlem1ssl  7246  recexprlem1ssu  7247  ltmprr  7255  mulcmpblnrlemg  7340  caucvgsrlemoffcau  7397  negsub  7784  neg2sub  7796  divmuleqap  8238  divneg2ap  8257  qapne  9178  binom2  10119  bcpasc  10228  crim  10346  remullem  10359  max0addsup  10706  isummolem2a  10825  isum1p  10940  geo2sum  10962  cvgratz  10980  efi4p  11062  tanaddap  11084  addcos  11091  cos2tsin  11096  demoivreALT  11117  omeo  11230  sqgcd  11350