Theorem 3eqtr3ri 2170

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 2163	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2163	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1332

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 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133

This theorem is referenced by:  indif2  3325  resdm2  5037  co01  5061  cocnvres  5071  undifdc  6820  1mhlfehlf  8962  rei  10703  resqrexlemover  10814  cos1bnd  11502  6gcd4e2  11719  3lcm2e6  11874  cosq23lt0  12962  sincos4thpi  12969  sincos6thpi  12971  cosq34lt1  12979