Theorem sn-iotauni 40180

Description: Version of iotauni 6403 using df-iota 6386 instead of dfiota2 6387. (Contributed by SN, 6-Nov-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotauni

Step	Hyp	Ref	Expression
1		iotavallem 40179	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		unieq 4852	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 3435	. . . . 5 ⊢ 𝑦 ∈ V
4	3	unisn 4863	. . . 4 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	eqtr2di 2795	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑})
6	1, 5	eqtrd 2778	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
7	6	exlimiv 1933	1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1782  {cab 2715  {csn 4563  ∪ cuni 4841  ℩cio 6384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3433  df-un 3893  df-in 3895  df-ss 3905  df-sn 4564  df-pr 4566  df-uni 4842  df-iota 6386

This theorem is referenced by:  sn-iotassuni  40183