Theorem 3eqtr3rd 2394

Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)

Hypotheses

Ref	Expression
3eqtr3d.1	|- (ph -> A = B)
3eqtr3d.2	|- (ph -> A = C)
3eqtr3d.3	|- (ph -> B = D)

Assertion

Ref	Expression
3eqtr3rd	|- (ph -> D = C)

Proof of Theorem 3eqtr3rd

Step	Hyp	Ref	Expression
1		3eqtr3d.3	. 2 |- (ph -> B = D)
2		3eqtr3d.1	. . 3 |- (ph -> A = B)
3		3eqtr3d.2	. . 3 |- (ph -> A = C)
4	2, 3	eqtr3d 2387	. 2 |- (ph -> B = C)
5	1, 4	eqtr3d 2387	1 |- (ph -> D = C)

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: (None)