Theorem dfaiota3 46372

Description: Alternate definition of ℩': this is to df-aiota 46365 what dfiota4 6529 is to df-iota 6489. operation using the if operator. It is simpler than df-aiota 46365 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 46371	. 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
2		aiotavb 46370	. . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
3	2	biimpi 215	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4		ifval 4565	. 2 ⊢ ((℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
5	1, 3, 4	mpbir2an 708	1 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533  ∃!weu 2556  {cab 2703  Vcvv 3468  ifcif 4523  ∩ cint 4943  ℩'caiota 46363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-uni 4903  df-int 4944  df-iota 6489  df-aiota 46365

This theorem is referenced by: (None)