Theorem 3eqtr4a 2147

Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtr4a.1	|- A = B
3eqtr4a.2	|- ( ph -> C = A )
3eqtr4a.3	|- ( ph -> D = B )

Assertion

Ref	Expression
3eqtr4a	|- ( ph -> C = D )

Proof of Theorem 3eqtr4a

Step	Hyp	Ref	Expression
1		3eqtr4a.2	. . 3 |- ( ph -> C = A )
2		3eqtr4a.1	. . 3 |- A = B
3	1, 2	syl6eq 2137	. 2 |- ( ph -> C = B )
4		3eqtr4a.3	. 2 |- ( ph -> D = B )
5	3, 4	eqtr4d 2124	1 |- ( ph -> C = 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:  uniintsnr  3732  fndmdifcom  5421  offres  5922  1stval2  5942  2ndval2  5943  ecovcom  6415  ecovass  6417  ecovdi  6419  zeo  8914  xnegneg  9358  fzsuc2  9556  expnegap0  10026  facp1  10201  bcpasc  10237  hashfzp1  10295  resunimafz0  10299  absexp  10575  fsumf1o  10845  fsumadd  10863  fisumrev2  10903  fsumparts  10927  efexp  11035  tanval2ap  11067  gcdcom  11306  gcd0id  11311  dfgcd3  11340  gcdass  11345  lcmcom  11387  lcmneg  11397  lcmass  11408  sqrt2irrlem  11481  nn0gcdsq  11519  dfphi2  11537  setscom  11597