Theorem 3eqtr3ri 2264

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 2257	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2257	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1398

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  indif2  3469  resdm2  5258  co01  5282  cocnvres  5292  undifdc  7197  1mhlfehlf  9473  rei  11609  resqrexlemover  11720  cos1bnd  12470  m1bits  12671  6gcd4e2  12716  3lcm2e6  12882  karatsuba  13153  ballotfilemth  13225  cosq23lt0  15824  sincos4thpi  15831  sincos6thpi  15833  cosq34lt1  15841