Theorem 3eqtrrd 2390

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtrd.1	⊢ (φ → A = B)
3eqtrd.2	⊢ (φ → B = C)
3eqtrd.3	⊢ (φ → C = D)

Assertion

Ref	Expression
3eqtrrd	⊢ (φ → D = A)

Proof of Theorem 3eqtrrd

Step	Hyp	Ref	Expression
1		3eqtrd.1	. . 3 ⊢ (φ → A = B)
2		3eqtrd.2	. . 3 ⊢ (φ → B = C)
3	1, 2	eqtrd 2385	. 2 ⊢ (φ → A = C)
4		3eqtrd.3	. 2 ⊢ (φ → C = D)
5	3, 4	eqtr2d 2386	1 ⊢ (φ → D = A)

This proof depends on syntax axioms:   → wi 4   = wceq 1642

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

This theorem is used by: (None)