Theorem iotauni2 6542

Description: Version of iotauni 6548 using df-iota 6525 instead of dfiota2 6526. (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 6541	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		unieq 4942	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		unisnv 4951	. . . 4 ⊢ ∪ {𝑦} = 𝑦
4	2, 3	eqtr2di 2797	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑})
5	1, 4	eqtrd 2780	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	5	exlimiv 1929	1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1537  ∃wex 1777  {cab 2717  {csn 4648  ∪ cuni 4931  ℩cio 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-sn 4649  df-pr 4651  df-uni 4932  df-iota 6525

This theorem is referenced by:  iotassuni  6545