Theorem iotauni 5901

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 2502	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		iotaval 5900	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3		uniabio 5899	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑥 ∣ 𝜑} = 𝑧)
4	2, 3	eqtr4d 2688	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
5	4	exlimiv 1898	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	1, 5	sylbi 207	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521   = wceq 1523  ∃wex 1744  ∃!weu 2498  {cab 2637  ∪ cuni 4468  ℩cio 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-sbc 3469  df-un 3612  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889

This theorem is referenced by:  iotaint  5902  iotassuni  5905  dfiota4  5917  dfiota4OLD  5918  fveu  6221  riotauni  6657