Theorem dfaiota3 47988

Description: Alternate definition of ℩', using the if operator: this is to df-aiota 47981 what dfiota4 6529 is to df-iota 6493. It is simpler than df-aiota 47981 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 47987	. 2 ⊢ (∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
2		aiotavb 47986	. . 3 ⊢ (¬ ∃!𝑥𝜑 ↔ (℩'𝑥𝜑) = V)
3	2	biimpi 219	. 2 ⊢ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)
4		ifval 4528	. 2 ⊢ ((℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ ((∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑}) ∧ (¬ ∃!𝑥𝜑 → (℩'𝑥𝜑) = V)))
5	1, 3, 4	mpbir2an 724	1 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃!weu 2595  {cab 2740  Vcvv 3453  ifcif 4485  ∩ cint 4910  ℩'caiota 47979

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-uni 4871  df-int 4911  df-iota 6493  df-aiota 47981

This theorem is used by: (None)