Theorem iotauni2 6510

Description: Version of iotauni 6515 using df-iota 6494 instead of dfiota2 6495. (Contributed by SN, 6-Nov-2024.)

Assertion

Ref	Expression
iotauni2	⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotauni2

Step	Hyp	Ref	Expression
1		iotaval2 6509	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		unieq 4877	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		unisnv 4886	. . . 4 ⊢ ∪ {𝑦} = 𝑦
4	2, 3	eqtr2di 2812	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑})
5	1, 4	eqtrd 2795	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	5	exlimiv 1962	1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∃wex 1811  {cab 2738  {csn 4584  ∪ cuni 4866  ℩cio 6492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6494

This theorem is used by:  iotassuni  6513