Theorem 3eqtr4a 2098

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 2088	. 2 |- ( ph -> C = B )
4		3eqtr4a.3	. 2 |- ( ph -> D = B )
5	3, 4	eqtr4d 2075	1 |- ( ph -> C = D )

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:  uniintsnr  3651  fndmdifcom  5273  offres  5762  1stval2  5782  2ndval2  5783  ecovcom  6213  ecovass  6215  ecovdi  6217  zeo  8343  xnegneg  8746  fzsuc2  8941  expnegap0  9263  absexp  9675  sqr2irrlem  9877