Theorem 3eqtrri 2232

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 2227	. 2 ⊢ 𝐴 = 𝐶
4		3eqtri.3	. 2 ⊢ 𝐶 = 𝐷
5	3, 4	eqtr2i 2228	1 ⊢ 𝐷 = 𝐴

Syntax hints:   = wceq 1373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-cleq 2199

This theorem is referenced by:  resindm  5010  dfdm2  5226  cofunex2g  6208  df1st2  6318  df2nd2  6319  enq0enq  7564  dfn2  9328  9p1e10  9526  0.999...  11907  pockthi  12756  sincosq3sgn  15375  sincosq4sgn  15376