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 1398

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

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

This theorem is referenced by:  indif2  3453  resdm2  5234  co01  5258  cocnvres  5268  undifdc  7159  1mhlfehlf  9404  rei  11522  resqrexlemover  11633  cos1bnd  12383  m1bits  12584  6gcd4e2  12629  3lcm2e6  12795  karatsuba  13066  cosq23lt0  15627  sincos4thpi  15634  sincos6thpi  15636  cosq34lt1  15644