See, that's where I don't think you're seeing the hidden arguments. BASIC programming jokes aside, the argument is:
"I believe life starts at conception."
"I don't believe life starts at conception."
The reasoning for these conclusions come from different places, but they both exist.
You don't hear a lot of pro-choice people explicitly say this, because there's really no point to it. "Women's rights" isn't used because there's no response available; it's used to shift context and reframe. If one side tries to gain a moral advantage by insisting life begins at conception, the other side shifts the argument away from the nebulous religious and biological arguments, and reframes it as a woman's right -- that way, the other side feels has to assume (or admit) an anti-woman point of view, losing a portion of moral ground.
But you know what? This is just another man talking about abortion. I'll just re-quote QG.
The pro-choice people I know who were raised pro-life were converted by arguments of reducing abortions through education and contraceptive availability, and from there walking back on late term bans (since most late term abortions are non-viable fetuses and/or threatening the life of the mother), and eventually walking back on all abortion access.
Compassion sells, not science.
I swear to God(dess) if anyone with a penis WELL ACTUALLYs me on this one they will wake up short one testicle.