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  5218  co01  5242  cocnvres  5252  undifdc  7082  1mhlfehlf  9325  rei  11405  resqrexlemover  11516  cos1bnd  12265  m1bits  12466  6gcd4e2  12511  3lcm2e6  12677  karatsuba  12948  cosq23lt0  15501  sincos4thpi  15508  sincos6thpi  15510  cosq34lt1  15518