Dendrodendritic synapse through axodendritic synapse at same dendrite?
I'm not sure if you are asking if an axon that sends a signal to a dendrite would then be released from that dendrite to another dendrite?

I don't see how this is possible seeing as the dendrite is already paired with the axon and downstream of said dendrite there would be a neuron.

A dendrodenritic synapse is two dendrites of two different neurons in contact with each other. For there to be a signal an action potential must have been reached.

Synapse is a junction between two neurons where chemical signaling occurs, but it is not the definition of a signal occurring. If that makes sense.

It seems like you are confused on the location and anatomy of a synapse.

There wouldn't be an axodendritic synapse followed by a dendrodendritic synapse then the neuron. It would either be axodendritic or dendrodendritic. Now there can be multitudes of connections between neurons but I don't think that is what you were asking.

Edit: Seeing your edit makes your question much clearer.

If the signal from the axodendritic synapse excites the neuron enough to reach action potential then that signal would be sent out through its synapses, including the dendrodendritic ones. So I guess in this way it was a "consequence" of the axodendritic synapse.

Though I would rather you think about it in that the action potential of one neuron sent a signal through it's connections to the neighborhing neuron, thus causing that neuron to reach action potential and further send out the signal. It's not really a "consequence" of the synapse, the synapse is more a pathway for the signal to travel.

The stimulus for the action potential is what causes a signal to fire and travel through a synapse.

Hopefully that answers your question.