Can I use Arduino Nano V3 to program another arduino (Pro Mini)?
Yes!

Yes, you can substitute the Arduino Nano V3 for the Arduino Uno and follow the instructions here, providing that the Arduino Pro Mini is a 5V version. (Otherwise, you will have to do some voltage conversion.)

Both boards have the same SPI pins as the Arduino Uno. The connections would be as follows:

Arduino Nano --> Arduino Pro Mini
5V  ------------------->  Vcc
GND  ---------------->  GND
D10  ----------------->  RESET (10uF between RESET &amp; GND)
D11  ----------------->  D11
D12  ----------------->  D12
D13  ----------------->  D13

The process detailed in the link can be summarized as follows:


Load the ArduinoISP sketch into the Arduino that you will use as programmer. You find it under Examples > 11. ArduinoISP > ArduinoISP. 
Select the serial port of the Arduino Nano (target): Tools > Port > corresponding port number.
Wire up the two Arduinos as described.
In the Tools menu, set the programmer to the Arduino as ISP.
Select the target board: Tools > Board > Arduino Pro Mini.
Open the program to be loaded to the Arduino Pro Mini and program it by selecting Sketch > Upload using programmer