Theorem dfiota4 6484

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 6469	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		iotanul 6472	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
3		ifval 4504	. 2 ⊢ ((℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) ↔ ((∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)))
4	1, 2, 3	mpbir2an 717	1 ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547  ∃!weu 2572  {cab 2718  ∅c0 4268  ifcif 4461  ∪ cuni 4845  ℩cio 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-uni 4846  df-iota 6448

This theorem is referenced by: (None)