Theorem 3eqtr3ri 2236

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 2229	. 2 ⊢ 𝐵 = 𝐶
5	1, 4	eqtr3i 2229	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1373

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-cleq 2199

This theorem is referenced by:  indif2  3421  resdm2  5187  co01  5211  cocnvres  5221  undifdc  7042  1mhlfehlf  9285  rei  11295  resqrexlemover  11406  cos1bnd  12155  m1bits  12356  6gcd4e2  12401  3lcm2e6  12567  karatsuba  12838  cosq23lt0  15390  sincos4thpi  15397  sincos6thpi  15399  cosq34lt1  15407