Theorem 3eqtr2ri 2205

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

Hypotheses

Ref	Expression
3eqtr2i.1	⊢ 𝐴 = 𝐵
3eqtr2i.2	⊢ 𝐶 = 𝐵
3eqtr2i.3	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
3eqtr2ri	⊢ 𝐷 = 𝐴

Proof of Theorem 3eqtr2ri

Step	Hyp	Ref	Expression
1		3eqtr2i.1	. . 3 ⊢ 𝐴 = 𝐵
2		3eqtr2i.2	. . 3 ⊢ 𝐶 = 𝐵
3	1, 2	eqtr4i 2201	. 2 ⊢ 𝐴 = 𝐶
4		3eqtr2i.3	. 2 ⊢ 𝐶 = 𝐷
5	3, 4	eqtr2i 2199	1 ⊢ 𝐷 = 𝐴

Syntax hints:   = wceq 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  funimacnv  5293  uniqs  6593  ef01bndlem  11764  cos2bnd  11768  sinhalfpilem  14215