Theorem 3eqtr3ri 2259

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 2252	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2252	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  indif2  3448  resdm2  5219  co01  5243  cocnvres  5253  undifdc  7097  1mhlfehlf  9340  rei  11425  resqrexlemover  11536  cos1bnd  12285  m1bits  12486  6gcd4e2  12531  3lcm2e6  12697  karatsuba  12968  cosq23lt0  15522  sincos4thpi  15529  sincos6thpi  15531  cosq34lt1  15539