Theorem sn-iotauni 40092

Description: Version of iotauni 6390 using df-iota 6373 instead of dfiota2 6374. (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		sn-iotaval 40091	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		unieq 4847	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 3427	. . . . 5 ⊢ 𝑦 ∈ V
4	3	unisn 4858	. . . 4 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	eqtr2di 2797	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑})
6	1, 5	eqtrd 2779	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
7	6	exlimiv 1938	1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1543  ∃wex 1787  {cab 2716  {csn 4558  ∪ cuni 4836  ℩cio 6371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-v 3425  df-un 3889  df-in 3891  df-ss 3901  df-sn 4559  df-pr 4561  df-uni 4837  df-iota 6373

This theorem is referenced by:  sn-iotassuni  40094