Theorem 3eqtr3ri 2167

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 2160	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2160	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1331

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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130

This theorem is referenced by:  indif2  3315  resdm2  5024  co01  5048  cocnvres  5058  undifdc  6805  1mhlfehlf  8931  rei  10664  resqrexlemover  10775  cos1bnd  11455  6gcd4e2  11672  3lcm2e6  11827  cosq23lt0  12903  sincos4thpi  12910  sincos6thpi  12912  cosq34lt1  12920