Theorem dfiota4 6530

Description: The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.)

Assertion

Ref	Expression
dfiota4	⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅)

Proof of Theorem dfiota4

Step	Hyp	Ref	Expression
1		iotauni 6515	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		iotanul 6518	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
3		ifval 4524	. 2 ⊢ ((℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)))
4	1, 2, 3	mpbir2an 724	1 ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569  ∃!weu 2593  {cab 2738  ∅c0 4278  ifcif 4481  ∪ cuni 4866  ℩cio 6492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6494

This theorem is used by: (None)