Theorem 3eqtr3ri 2207

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 2200	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2200	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  indif2  3381  resdm2  5121  co01  5145  cocnvres  5155  undifdc  6925  1mhlfehlf  9139  rei  10910  resqrexlemover  11021  cos1bnd  11769  6gcd4e2  11998  3lcm2e6  12162  cosq23lt0  14293  sincos4thpi  14300  sincos6thpi  14302  cosq34lt1  14310