If you think about it that would be completely illogical. The pawn isn't pinned because White's King would fall 1st.
Imagine that...
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Imagine that a pinned piece cannot check the opponent king. Take the diagram below as an example where the black pawn is pinned by the white rook. Although, the pawn is still able to move forward, it cannot capture diagonally because of that pin, theoretically speaking. Is it possible then to fancy a chess game with the same position where the white king is safe (no check) due to the disability of the pinned pawn?