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Theorem iota1 4353
 Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
iota1 (∃!xφ → (φ ↔ (℩xφ) = x))

Proof of Theorem iota1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2208 . 2 (∃!xφzx(φx = z))
2 sp 1747 . . . . 5 (x(φx = z) → (φx = z))
3 iotaval 4350 . . . . . 6 (x(φx = z) → (℩xφ) = z)
43eqeq2d 2364 . . . . 5 (x(φx = z) → (x = (℩xφ) ↔ x = z))
52, 4bitr4d 247 . . . 4 (x(φx = z) → (φx = (℩xφ)))
6 eqcom 2355 . . . 4 (x = (℩xφ) ↔ (℩xφ) = x)
75, 6syl6bb 252 . . 3 (x(φx = z) → (φ ↔ (℩xφ) = x))
87exlimiv 1634 . 2 (zx(φx = z) → (φ ↔ (℩xφ) = x))
91, 8sylbi 187 1 (∃!xφ → (φ ↔ (℩xφ) = x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204  ℩cio 4337 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-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  iota2df  4365  sniota  4369
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