Theorem 3eqtrri 2264

Description: An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtri.1	|- A = B
3eqtri.2	|- B = C
3eqtri.3	|- C = D

Assertion

Ref	Expression
3eqtrri	|- D = A

Proof of Theorem 3eqtrri

Step	Hyp	Ref	Expression
1		3eqtri.1	. . 3 |- A = B
2		3eqtri.2	. . 3 |- B = C
3	1, 2	eqtri 2259	. 2 |- A = C
4		3eqtri.3	. 2 |- C = D
5	3, 4	eqtr2i 2260	1 |- D = A

This proof depends on syntax axioms:   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  resindm  5105  dfdm2  5322  cofunex2g  6339  df1st2  6455  df2nd2  6456  enq0enq  7798  dfn2  9576  9p1e10  9779  0.999...  12288  pockthi  13137  sincosq3sgn  15929  sincosq4sgn  15930  0grsubgr  16505