Theorem iotauni 4352

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 2208	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
2		iotaval 4351	. . . 4 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z)
3		uniabio 4350	. . . 4 ⊢ (∀x(φ ↔ x = z) → ∪{x ∣ φ} = z)
4	2, 3	eqtr4d 2388	. . 3 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = ∪{x ∣ φ})
5	4	exlimiv 1634	. 2 ⊢ (∃z∀x(φ ↔ x = z) → (℩xφ) = ∪{x ∣ φ})
6	1, 5	sylbi 187	1 ⊢ (∃!xφ → (℩xφ) = ∪{x ∣ φ})

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  ∪cuni 3892  ℩cio 4338

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 2479  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  iotaint  4353  iotassuni  4356  dfiota3  4371  dfiota4  4373