Theorem iotauni2 6511

Description: Version of iotauni 6517 using df-iota 6495 instead of dfiota2 6496. (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 6510	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		unieq 4900	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		unisnv 4909	. . . 4 ⊢ ∪ {𝑦} = 𝑦
4	2, 3	eqtr2di 2786	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑})
5	1, 4	eqtrd 2769	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	5	exlimiv 1929	1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1778  {cab 2712  {csn 4608  ∪ cuni 4889  ℩cio 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3466  df-un 3938  df-ss 3950  df-sn 4609  df-pr 4611  df-uni 4890  df-iota 6495

This theorem is referenced by:  iotassuni  6514