Theorem iotauni 4822

Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iotauni	⊢ (∃!xφ → (℩xφ) = ∪ {x ∣ φ})

Proof of Theorem iotauni

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 1900	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
2		iotaval 4821	. . . 4 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z)
3		uniabio 4820	. . . 4 ⊢ (∀x(φ ↔ x = z) → ∪ {x ∣ φ} = z)
4	2, 3	eqtr4d 2072	. . 3 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = ∪ {x ∣ φ})
5	4	exlimiv 1486	. 2 ⊢ (∃z∀x(φ ↔ x = z) → (℩xφ) = ∪ {x ∣ φ})
6	1, 5	sylbi 114	1 ⊢ (∃!xφ → (℩xφ) = ∪ {x ∣ φ})

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378  ∃!weu 1897  {cab 2023  ∪ cuni 3571  ℩cio 4808

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810

This theorem is referenced by:  iotaint  4823  fveu  5113  riotauni  5417