Theorem dfaiota3 47041

Description: Alternate definition of ℩': this is to df-aiota 47034 what dfiota4 6554 is to df-iota 6515. operation using the if operator. It is simpler than df-aiota 47034 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 47040	. 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
2		aiotavb 47039	. . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
3	2	biimpi 216	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4		ifval 4572	. 2 ⊢ ((℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
5	1, 3, 4	mpbir2an 711	1 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1536  ∃!weu 2565  {cab 2711  Vcvv 3477  ifcif 4530  ∩ cint 4950  ℩'caiota 47032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-uni 4912  df-int 4951  df-iota 6515  df-aiota 47034

This theorem is referenced by: (None)