Theorem 3eqtr3i 2381

Description: An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtr3i.1	⊢ A = B
3eqtr3i.2	⊢ A = C
3eqtr3i.3	⊢ B = D

Assertion

Ref	Expression
3eqtr3i	⊢ C = D

Proof of Theorem 3eqtr3i

Step	Hyp	Ref	Expression
1		3eqtr3i.1	. . 3 ⊢ A = B
2		3eqtr3i.2	. . 3 ⊢ A = C
3	1, 2	eqtr3i 2375	. 2 ⊢ B = C
4		3eqtr3i.3	. 2 ⊢ B = D
5	3, 4	eqtr3i 2375	1 ⊢ C = D

Syntax hints:   = wceq 1642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346

This theorem is referenced by:  csbvarg  3164  un12  3422  in12  3467  indif1  3500  difundi  3508  difundir  3509  difindi  3510  difindir  3511  dif32  3518  undif1  3626  addc4  4418  addc6  4419  tfin1c  4500  xp0  5045  caov12  5637  caov13  5639  caov411  5641  caovdir  5643  tcdi  6165  tc2c  6167  ce0addcnnul  6180  nchoicelem1  6290