Theorem 3eqtr3ri 2129

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 2122	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2122	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1299

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-4 1455  ax-17 1474  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-cleq 2093

This theorem is referenced by:  indif2  3267  resdm2  4965  co01  4989  cocnvres  4999  undifdc  6741  1mhlfehlf  8790  rei  10512  resqrexlemover  10622  cos1bnd  11264  6gcd4e2  11476  3lcm2e6  11631