Theorem dfaiota3 47562

Description: Alternate definition of ℩', using the if operator: this is to df-aiota 47555 what dfiota4 6484 is to df-iota 6448. It is simpler than df-aiota 47555 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.)

Assertion

Ref	Expression
dfaiota3	⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)

Proof of Theorem dfaiota3

Step	Hyp	Ref	Expression
1		aiotaint 47561	. 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
2		aiotavb 47560	. . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
3	2	biimpi 217	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4		ifval 4504	. 2 ⊢ ((℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
5	1, 3, 4	mpbir2an 717	1 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547  ∃!weu 2572  {cab 2718  Vcvv 3432  ifcif 4461  ∩ cint 4884  ℩'caiota 47553

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  ax-sep 5225

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  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-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-uni 4846  df-int 4885  df-iota 6448  df-aiota 47555

This theorem is referenced by: (None)