How do I stream one thing to our online audience while a completely different program is unfolding in house?

If I stream video to our online audience while inhouse our pastor is ministering to our congregants, the online users will hear his mic overtop of the video.  We need to use the mic inhouse so the audience can hear him but how do I stop our online users from being able to hear him and yet still hear the videos that will be playing for them?

Send your EW output to a program like OBS, XSplit, etc. and send that programs output to your stream. Set up different Scenes (in OBS, not sure what other programs call it) for the different sources. One for EW, one for the videos, etc., then switch to be tween the scenes as necessary. You can setup as many scenes as you find necessary. Each scene can have it's own audio source so they don't overlap. 

Use an audio mixer to manage the audio sources. Connect the microphone used by your pastor to one channel of the mixer, and the audio from the video program to another channel. This will allow you to control the volume levels independently.