Theorem 3eqtr3ri 2268

Description: An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)

Hypotheses

Ref	Expression
3eqtr3i.1	⊢ 𝐴 = 𝐵
3eqtr3i.2	⊢ 𝐴 = 𝐶
3eqtr3i.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3eqtr3ri	⊢ 𝐷 = 𝐶

Proof of Theorem 3eqtr3ri

Step	Hyp	Ref	Expression
1		3eqtr3i.3	. 2 ⊢ 𝐵 = 𝐷
2		3eqtr3i.1	. . 3 ⊢ 𝐴 = 𝐵
3		3eqtr3i.2	. . 3 ⊢ 𝐴 = 𝐶
4	2, 3	eqtr3i 2261	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2261	1 ⊢ 𝐷 = 𝐶

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:  indif2  3475  resdm2  5281  co01  5305  cocnvres  5315  undifdc  7237  1mhlfehlf  9534  rei  11686  resqrexlemover  11797  cos1bnd  12550  m1bits  12751  6gcd4e2  12796  3lcm2e6  12963  karatsuba  13238  ballotfilemth  13338  cosq23lt0  16077  sincos4thpi  16084  sincos6thpi  16086  cosq34lt1  16095  cht3  16294  bclbnd  16324  bposlem8  16335