Theorem 3eqtr3ri 2195

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 2188	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2188	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  indif2  3366  resdm2  5094  co01  5118  cocnvres  5128  undifdc  6889  1mhlfehlf  9075  rei  10841  resqrexlemover  10952  cos1bnd  11700  6gcd4e2  11928  3lcm2e6  12092  cosq23lt0  13394  sincos4thpi  13401  sincos6thpi  13403  cosq34lt1  13411