A DEPARTING DC9-30 FLC GETS A TCASII RA FROM TFC CLBING TO THEIR R OUT OF PIT; PA.

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DEPARTED PIT ON RWY 28R; ASSIGNED 340 DEG HDG AND CLB TO 5000 FT. PASSING THROUGH AT 500 FT AGL STARTED TURN AND GIVEN NEW HEADING OF 360 DEGS AND SWITCH TO DEP CTL FREQ. DEP CTL ASSIGNED 110 DEG HDG AND CLB TO 14000 FT. ACFT WAS CLEANED UP ON SCHEDULE AND AFTER TKOF CHKLIST COMPLETED; CAPT SIMULTANEOUSLY NOTICED TA ON TCASII AND HEARD 'MONITOR VERT SPD' (PASSING THROUGH 80 DEG HDG AND AT 3000 FT AGL). INITIAL TA WAS 3 O'CLOCK POS AND WITHIN 2 MI ON TCASII SCREEN. WITHIN SECONDS (2-3) AN RA ON TCASII AND 'DSND; DSND;' FOLLOWED IMMEDIATELY WITH 'CLB NOW.' CAPT CLBED AT MAX CLB (AT 2000 FPM) AND TURNED AWAY FROM TFC AT 60 DEG HDG. WHILE FO TRIED UNSUCCESSFULLY TO CONTACT TFC VISUALLY AND CONFIRM TFC WITH DEP CTL; DEP CTL GAVE ONE CLRNC TO 'INCREASE CLB RATE.' CLRED TFC AND CONTINUED UNEVENTFULLY TO DEST. CALLED ATC AFTER ARRIVING AT DEST AND TOLD THAT TWR HAD ISSUED A TKOF TO ACFT #2 IMMEDIATELY AFTER OUR TKOF AND GIVEN A HDG OF 360 DEGS. ATC SUPVR ADMITTED ERROR ON ATC AND TOLD US 2 CTLRS HAD BEEN RELIEVED OF DUTY AND THAT ONE HAD BEEN DECERTIFIED BECAUSE OF THE INCIDENT. INCIDENT RPTED TO COMPANY VIA SAFETY RPT AND TELEPHONE CONVERSATIONS. ATC FOLLOWING DAY INDICATED NEAR MISS AT LESS THAN .8 MI HORIZ AND 0 FT VERT. ALTHOUGH TCASII WARNING WAS MINIMAL; QUICK REACTIONS BY CREW AND PROPER TCASII COMMANDS PREVENTED CATASTROPHIC INCIDENT.