Theorem 3eqtr4d 2395

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtr4d.1	⊢ (φ → A = B)
3eqtr4d.2	⊢ (φ → C = A)
3eqtr4d.3	⊢ (φ → D = B)

Assertion

Ref	Expression
3eqtr4d	⊢ (φ → C = D)

Proof of Theorem 3eqtr4d

Step	Hyp	Ref	Expression
1		3eqtr4d.2	. 2 ⊢ (φ → C = A)
2		3eqtr4d.3	. . 3 ⊢ (φ → D = B)
3		3eqtr4d.1	. . 3 ⊢ (φ → A = B)
4	2, 3	eqtr4d 2388	. 2 ⊢ (φ → D = A)
5	1, 4	eqtr4d 2388	1 ⊢ (φ → C = D)

Syntax hints:   → wi 4   = 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:  addcdir  6252