Theorem iotauni 6506

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		eu6 2573	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		iotaval 6502	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3		uniabio 6498	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑥 ∣ 𝜑} = 𝑧)
4	2, 3	eqtr4d 2773	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
5	4	exlimiv 1930	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	1, 5	sylbi 217	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779  ∃!weu 2567  {cab 2713  ∪ cuni 4883  ℩cio 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-sn 4602  df-pr 4604  df-uni 4884  df-iota 6484

This theorem is referenced by:  iotaint  6507  iotassuniOLD  6510  dfiota4  6523  fveu  6865  riotauni  7368  afv2eu  47267