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From Wikipedia, citing Dr. Cavanaugh:

This set of criteria states that an action having foreseen harmful effects practically inseparable from the good effect is justifiable if the following are true:

the nature of the act is itself good, or at least morally neutral;
the agent intends the good effect and not the bad either as a means to the good or as an end itself;
the good effect outweighs the bad effect in circumstances sufficiently grave to justify causing the bad effect and the agent exercises due diligence to minimize the harm.

Principle of double effect

Of interest to us is that the first two are separate. There is the intrinsic nature of the act, and the intent of the act. You are correct that the intent of the firefighter is good. But that speaks nothing of the intrinsic nature of the act.

The intrinsic nature of pushing a woman down the stairs is to cause an injury. It is in the nature of falling down from a certain height that an injury results. That is itself does not yet fully describe the act: for example, similarly injuring the enemy in a just war is not intrinsically bad. However, the woman is not an enemy: her injury is not preventing her from doing harm. It is then intrinsically bad act of causing injury to an innocent.

If that does not work to you, please explain how anything can be intrinsically bad in your system. For example, how is it that abortion is intrinsically bad but causing the same fatal injury to an enemy soldier is not bad.

Annalex, this is a really good discussion, and only reinforces to me that I need to engage this kind of analysis more often, and with similarly well-informed persons as yourself.

I can't answer your hhouht-provoking questions right now, because Ih ave to get at my garden again before it rains. You know how it is in the Springtime, weeks of enforced idleness because the weather isn't right, and then all of a sudden everything is urgent, and needs to be done at once!

I WILL get back later today. If I don't, give me a sharp kick!