FLC OF A GULFSTREAM; G4; OVERSHOT ASSIGNED CLB ALT XING RESTR ON A SID DEP RESULTING IN LTSS WITH ANOTHER ACFT.

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JUST BEFORE HOLDING POINT C1 I WAS ISSUED A CLRNC AND READ BACK WE WERE CLRED DAVENTRY 1B. GND SAID CONTACT TWR. I BRIEFED THE PF AND I READ THE CLRNC FROM L TO R; IE; AT D20 DTY 6000 FT D25 ABOVE 4500 FT D30 DTY ABOVE 3500 FT AND AT D24 LON AT OR ABOVE 4000 FT; AND I READ THE REST OF THE SID TO THE PF. MY EYES READ ABOVE AND WHEN I GOT TO D24 LON IT WAS AT OR BELOW AND NOT ABOVE. I PUT 6000 FT IN THE ALT SELECT BECAUSE THAT WAS THE HIGHEST ALT ON THE SID. I SELECTED THE SID ON THE FMS AND REVIEWED IT TO COMPARE WITH THE SID AND IT WAS CORRECT. I CONTACTED THE TWR FOR TKOF. WE WERE CLRED ONTO THE RWY AND BACK TRACK INTO POS ON RWY 26. AS WE COMPLETED THE TURN THE TWR CLRED US FOR TKOF. AT 500 FT AGL THE PF ENGAGED THE LNAV BUT VNAV WAS NOT SELECTED INADVERTENTLY WHICH CAUSED THE AUTOPLT TO FLY TO 6000 FT IMMEDIATELY AND NOT LEVEL OFF AT 4000 FT AS REQUIRED BY THE SID. THE PF CALLED FOR THE AFTER TKOF CHKLIST AND I WAS COMPLETING IT WHEN THE CTLR CALLED AND ASKED WHAT ALT WE WERE AT AND I LOOKED UP FROM DOING THE CHKLIST AND REPLIED THAT WE WERE AT 6000 FT. HE SAID THE SID SAID WE WERE TO BE AT 4000 FT PER THE SID AND I SAID THE SID SAID 'AT OR ABOVE.' HE ASKED ME TO READ THE SID AGAIN; ONLY TO FIND OUT FROM THE NAV DISPLAY THAT WE HAD NOT IN FACT PASSED THE D24 LON 015 DEG RADIAL POINT YET AND THE SID ACTUALLY SAID 'AT OR BELOW.' I RESPONDED THAT HE WAS CORRECT AND HE TOLD US TO DSND TO 4000 FT. (THE PF WAS ALREADY DSNDING PRIOR TO HIS COMMAND.) HE SAID I WAS INVOLVED IN A NEAR MISS AND HE WAS FILING A RPT WITH LONDON CTL. THE TCASII NEVER HAD A RA NOR INDICATED AN ACFT NEAR US. THE CTLR DID SAY AN ACFT PASSED OVERHEAD AT FL70 AND WE WERE AT 6000 FT ON A QNH OF 1020 MILLIBARS.