Theorem dfaiota3 47216

Description: Alternate definition of ℩': this is to df-aiota 47209 what dfiota4 6478 is to df-iota 6442. operation using the if operator. It is simpler than df-aiota 47209 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 47215	. 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
2		aiotavb 47214	. . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
3	2	biimpi 216	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4		ifval 4517	. 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 1541  ∃!weu 2565  {cab 2711  Vcvv 3437  ifcif 4474  ∩ cint 4897  ℩'caiota 47207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-uni 4859  df-int 4898  df-iota 6442  df-aiota 47209

This theorem is referenced by: (None)