Theorem iotauni 5780

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 2462	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		iotaval 5779	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
3		uniabio 5778	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑥 ∣ 𝜑} = 𝑧)
4	2, 3	eqtr4d 2647	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
5	4	exlimiv 1845	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	1, 5	sylbi 206	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695  ∃!weu 2458  {cab 2596  ∪ cuni 4372  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768

This theorem is referenced by:  iotaint  5781  iotassuni  5784  dfiota4  5796  fveu  6095  riotauni  6517