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Theorem tceq 6158
Description: Equality theorem for cardinal T operator. (Contributed by SF, 2-Mar-2015.)
Assertion
Ref Expression
tceq (A = BTc A = Tc B)

Proof of Theorem tceq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2808 . . . 4 (A = B → (y A x = Nc 1yy B x = Nc 1y))
21anbi2d 684 . . 3 (A = B → ((x NC y A x = Nc 1y) ↔ (x NC y B x = Nc 1y)))
32iotabidv 4360 . 2 (A = B → (℩x(x NC y A x = Nc 1y)) = (℩x(x NC y B x = Nc 1y)))
4 df-tc 6103 . 2 Tc A = (℩x(x NC y A x = Nc 1y))
5 df-tc 6103 . 2 Tc B = (℩x(x NC y B x = Nc 1y))
63, 4, 53eqtr4g 2410 1 (A = BTc A = Tc B)
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358   = wceq 1642   wcel 1710  wrex 2615  1cpw1 4135  cio 4337   NC cncs 6088   Nc cnc 6091   Tc ctc 6093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-uni 3892  df-iota 4339  df-tc 6103
This theorem is referenced by:  tcdi  6164  tc2c  6166  tc11  6227  taddc  6228  tlecg  6229  letc  6230  ce0t  6231  ce2le  6232  cet  6233  tce2  6235  te0c  6236  ce0lenc1  6238  tlenc1c  6239  brtcfn  6245  nmembers1lem1  6267  nmembers1  6270  nchoicelem1  6287  nchoicelem2  6288  nchoicelem12  6298  nchoicelem16  6302  nchoicelem17  6303  nchoicelem19  6305  nchoice  6306
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