CTLR CLBED LGT THROUGH OCCUPIED ALT WITH LTSS. CONFLICT ALERT ACTIVATED AS WELL AS TCASII TA RA.

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THIS INCIDENT HAPPENED AT APPROX XX54 EST WHILE CLBING OUT OF ATL ENRTE TO LGA. WE WERE APPROX 35 MI SW OF THE SPA VOR AND HAD BEEN CLRED TO FL250. WE WERE PASSING THROUGH FL23.7 WHEN WE GOT A TFC ALERT ON OUR TCASII SYS WHICH INDICATED AN ACFT AT OUR 10 O'CLOCK POS COMING TOWARD US. ALMOST IMMEDIATELY; THE CTLR ASKED US IF WE WERE LEVEL AT FL230. WE REPLIED THAT WE HAD BEEN CLRED TO FL250 AT WHICH TIME HE TOLD US TO 'EXPEDITE' TO FL250; AND TOLD THE ACFT IN CONFLICT TO DSND. IN THE MEANTIME; I HAD SPOTTED THE TFC AND HAD TAKEN THE ACFT OFF AUTOPLT. I PULLED US ABRUPTLY WHILE KEEPING THE OTHER ACFT IN SIGHT AND INADVERTENTLY CLBED TO FL25.8 BEFORE LEVELING OFF. THE CTLR THEN ASKED US WHAT OUR ALT WAS AND WE REPLIED THAT WE HAD INADVERTENTLY CLBED TO FL25.8 IN RESPONSE TO THE TCASII ALERT AND WHILE VISUALLY TRYING TO AVOID THE CONFLICTING ACFT. HE THEN TOLD US TO CONTINUE CLB TO FL270. WE EXPLAINED TO THE CTLR THAT WE HAD INDEED BEEN CLRED TO FL250 AND TO CHK HIS TAPES. LATER AFTER BEING HANDED OFF TO THE NEXT CTLR; SHE RELAYED A MESSAGE FROM THE PREVIOUS CTLR SAYING THAT HE APPRECIATED THE EXPEDITIOUS CLB AND THAT THERE WAS NO PROBLEM ABOUT THE ALT. I BELIEVE THE PROBLEM OCCURRED BECAUSE THE CTLR HAD INITIALLY CLRED US TO FL230 AND THEN CHANGED HIS MIND AND TOLD US TO CONTINUE OUR CLB TO FL250. I THINK HE SIMPLY FORGOT THAT HE HAD CLRED US ON UP AND THAT RESULTED IN THE TFC CONFLICT. SUPPLEMENTAL INFO FROM ACN 193887. I BELIEVE THE PROBLEM STARTED WHEN ATL CENTER CLRED US AS FOLLOWS: 'CLRED TO FL230; EXPECT FL250 IN ...NEVER MIND; CLB TO FL250 THIS TIME.' OUR RESPONSE WAS 'ROGER; OUT OF FL220 FOR FL250.' I THINK THE CTLR JUST FORGOT THE ALT WE WERE CLRED TO.