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Theorem tfineq 4488
 Description: Equality theorem for the finite T operator. (Contributed by SF, 24-Jan-2015.)
Assertion
Ref Expression
tfineq (A = BTfin A = Tfin B)

Proof of Theorem tfineq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . 3 (A = B → (A = B = ))
2 rexeq 2808 . . . . 5 (A = B → (y A 1y xy B 1y x))
32anbi2d 684 . . . 4 (A = B → ((x Nn y A 1y x) ↔ (x Nn y B 1y x)))
43iotabidv 4360 . . 3 (A = B → (℩x(x Nn y A 1y x)) = (℩x(x Nn y B 1y x)))
51, 4ifbieq2d 3682 . 2 (A = B → if(A = , , (℩x(x Nn y A 1y x))) = if(B = , , (℩x(x Nn y B 1y x))))
6 df-tfin 4443 . 2 Tfin A = if(A = , , (℩x(x Nn y A 1y x)))
7 df-tfin 4443 . 2 Tfin B = if(B = , , (℩x(x Nn y B 1y x)))
85, 6, 73eqtr4g 2410 1 (A = BTfin A = Tfin B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∅c0 3550   ifcif 3662  ℘1cpw1 4135  ℩cio 4337   Nn cnnc 4373   Tfin ctfin 4435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-nan 1288  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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663  df-uni 3892  df-iota 4339  df-tfin 4443 This theorem is referenced by:  tfincl  4492  tfin11  4493  tfin1c  4499  tfinltfinlem1  4500  tfinltfin  4501  eventfin  4517  oddtfin  4518  sfintfin  4532  tfinnn  4534  vfinncvntnn  4548  vfinspsslem1  4550  vfinspss  4551  vfinspclt  4552
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