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  3449  resdm2  5223  co01  5247  cocnvres  5257  undifdc  7107  1mhlfehlf  9350  rei  11447  resqrexlemover  11558  cos1bnd  12307  m1bits  12508  6gcd4e2  12553  3lcm2e6  12719  karatsuba  12990  cosq23lt0  15544  sincos4thpi  15551  sincos6thpi  15553  cosq34lt1  15561