Jamieson, Fausset and Brown on Philippians 4:7

7. And--The inseparable consequence of thus laying everything before God in "prayer with thanksgiving."

peace--the dispeller of "anxious care" (Php 4:6).

of God--coming from God, and resting in God (Joh 14:27; 16:33; Col 3:15).

passeth--surpasseth, or exceedeth, all man's notional powers of understanding its full blessedness (1Co 2:9, 10; Ep 3:20; compare Pr 3:17).

shall keep--rather, "shall guard"; shall keep as a well-garrisoned stronghold (Is 26:1, 3). The same Greek verb is used in 1Pe 1:5. There shall be peace secure within, whatever outward troubles may besiege.

hearts and minds--rather, "hearts (the seat of the thoughts) and thoughts" or purposes.

through--rather as Greek, "in Christ Jesus." It is in Christ that we are "kept" or "guarded" secure.