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Theorem dfiota3 4370
 Description: The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
dfiota3 (℩xφ) = if(∃!xφ, {x φ}, )

Proof of Theorem dfiota3
StepHypRef Expression
1 iotauni 4351 . . 3 (∃!xφ → (℩xφ) = {x φ})
2 iftrue 3668 . . 3 (∃!xφ → if(∃!xφ, {x φ}, ) = {x φ})
31, 2eqtr4d 2388 . 2 (∃!xφ → (℩xφ) = if(∃!xφ, {x φ}, ))
4 iotanul 4354 . . 3 ∃!xφ → (℩xφ) = )
5 iffalse 3669 . . 3 ∃!xφ → if(∃!xφ, {x φ}, ) = )
64, 5eqtr4d 2388 . 2 ∃!xφ → (℩xφ) = if(∃!xφ, {x φ}, ))
73, 6pm2.61i 156 1 (℩xφ) = if(∃!xφ, {x φ}, )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642  ∃!weu 2204  {cab 2339  ∅c0 3550   ifcif 3662  ∪cuni 3891  ℩cio 4337 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-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by: (None)
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