B767 FLC INITIATE EVASIVE TCASII MANEUVER DUE TO OPPOSITE DIRECTION TFC. CLRNC DISPUTED BY ATC. AFTER REVIEW; ATC ACKNOWLEDGED CLRNC INCORRECTLY ASSIGNED WRONG ALT.

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I WAS PIC AND GIVING IOE TO A NEW FO. HE WAS THE PF. WE HAD BEEN GIVEN CLB TO 3000 FT AND RWY HDG ON INITIAL CLRNC. THE DEP CTLR ISSUED A TURN TO THE L OF 160 DEGS (TO 290 DEGS MAGNETIC) TO INTERCEPT THE DEP RADIAL AND CLB TO 7000 FT. I ACKNOWLEDGED BOTH AND RESET THE ALT REMINDER. THE FO ACKNOWLEDGED THE CHANGES VERBALLY AND POINTED TO THE ALT REMINDER (COMPANY POLICY). PASSING 5000 FT; I NOTICED TFC AHEAD AT 7000 FT ON THE TCASII WITH A 'MONITOR CLB' ANNOUNCEMENT. THEN AT 5600 FT; THE TARGET TURNED RED AND THE RA WENT TO DECREASE CLB WITH A VERY SMALL GREEN ARC ON THE IVSI. I ENUNCIATED THIS AND PUSHED THE YOKE TO THE GREEN ARC ON THE IVSI. THE CTLR ASKED OUR ALT AND I RESPONDED THAT WE WERE LEVELING AT 5600 FT AND RESPONDING TO AN RA TCASII WARNING. HE SAID TO MAINTAIN 5000 FT AND WE RETURNED TO 5000 FT. I ASKED WHAT CAUSED THE PROB AND HE SAID WE HAD ONLY BEEN CLRED TO 5000 FT; NOT 7000 FT. I SAID WE HAD HEARD AND READ BACK 7000 FT. 'WE'LL HAVE TO PLAY THE TAPES;' HE SAID. 10 MINS LATER WE WERE INSTRUCTED TO CALL HIM BACK AND WE DID. HE SAID THE TAPES HAD BEEN PLAYED AND WE 'HAD' BEEN CLRED TO 7000 FT AND THE MISTAKE WAS HIS; NOT OURS. IT SEEMS THE LATEST COURT RULINGS AGAINST PLTS IN 'READBACK/HEARBACK' CASES HAVE ALLOWED CTLRS TO IGNORE ALL OUR READBACKS. THIS RESPONSIBILITY SHOULD BE SHARED BTWN CTLRS AND PLTS. THE OTHER ACFT WAS A B737.