Quick, possibly dumb question

When it comes to recording gameplay in OBS, I know about the PS3 needing an HDMI splitter in order for the signal to go through, but I’m curious about the PSPlus streaming through the Classics catalogue. If I streamed PS3 games through the PS5, I should be able to capture the gameplay on OBS without a splitter correct? Sorry if it’s an inane question, but I never can seem to get a straight answer from research