Theorem 3eqtr3ri 2234

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 2227	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2227	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1372

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 1469  ax-gen 1471  ax-4 1532  ax-17 1548  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197

This theorem is referenced by:  indif2  3416  resdm2  5170  co01  5194  cocnvres  5204  undifdc  7003  1mhlfehlf  9237  rei  11129  resqrexlemover  11240  cos1bnd  11989  m1bits  12190  6gcd4e2  12235  3lcm2e6  12401  karatsuba  12672  cosq23lt0  15223  sincos4thpi  15230  sincos6thpi  15232  cosq34lt1  15240