A B733 CREW AFTER LNDG AT MDW MISINTERPED TAXI CLRNC AND CROSSED A RWY NOT GIVEN CLRNC TO CROSS.

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AFTER LNDG AND EXITING RWY 31C; WE WERE TOLD BY THE TWR CTLR TO 'CROSS RWY 31R; TAXI VIA TXWY N AND TXWY P; HOLD SHORT RWY 22L; CONTACT GND.' THE CLRNC WAS READ BACK AND WE SWITCHED TO GND CTL. GND INSTRUCTED US TO TAXI VIA TXWY N; TXWY P; CROSS RWY 22L. THE FO READ BACK THE CLRNC. I REPEATED THE CLRNC TO THE FO AND WE CONTINUED OUR TAXI. AS WE CROSSED RWY 22L THE GND CTLR ADVISED US THAT WE WERE TOLD TO HOLD SHORT OF RWY 22L. THERE WAS NO TFC ON THE RWY OR APCH. THE FO TOLD THE CTLR THAT HE (THE FO) HAD READ BACK A CLRNC TO CROSS RWY 22L. GND THEN SAID THAT IT WAS 'OK; THERE'S NO PROB.' WE CONTINUED OUR TAXI TO THE GATE. I THINK THAT SOME OF THE CONTRIBUTING FACTORS WERE: 1) THE LACK OF A READBACK OF THE HOLD SHORT INSTRUCTION BY THE FO. HE INSTEAD READ BACK THE CLRNC THAT WE BOTH HEARD. 2) THE GND CTLR NOT REQUESTING THE REQUIRED RWY HOLD SHORT READBACK. 3) THE FACT THAT NO MENTION OF RWY 22R WAS EVER MADE BY GND CTL; EVEN THOUGH IT MUST BE CROSSED IN ORDER TO REACH RWY 22L. 4) THE GND CTLR WAS BUSY WITH NUMEROUS ACFT THAT WERE WAITING ON TXWYS FOR THEIR GATES. THE GATES AND RAMPS WERE CONGESTED WITH ACFT BEING DEICED. SUPPLEMENTAL INFO FROM ACN 496937: I KNOW I HEARD TAXI T OTHER GATE AND IT IS WHAT I READ BACK. IF IT WASN'T WHAT THE GND CTLR SAID; HE SHOULD HAVE CORRECTED ME.