Theorem 3eqtr3ri 2261

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 2254	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2254	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1397

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  indif2  3451  resdm2  5227  co01  5251  cocnvres  5261  undifdc  7116  1mhlfehlf  9362  rei  11477  resqrexlemover  11588  cos1bnd  12338  m1bits  12539  6gcd4e2  12584  3lcm2e6  12750  karatsuba  13021  cosq23lt0  15576  sincos4thpi  15583  sincos6thpi  15585  cosq34lt1  15593