Theorem iotaval2 6499

Description: Version of iotaval 6502 using df-iota 6484 instead of dfiota2 6485. (Contributed by SN, 6-Nov-2024.)

Assertion

Ref	Expression
iotaval2	⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotaval2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iota 6484	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		eqeq1 2739	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤}))
3		sneqbg 4819	. . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤))
4	3	elv 3464	. . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)
5		equcom 2017	. . . . . 6 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
6	4, 5	bitri 275	. . . . 5 ⊢ ({𝑦} = {𝑤} ↔ 𝑤 = 𝑦)
7	2, 6	bitrdi 287	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
8	7	alrimiv 1927	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
9		uniabio 6498	. . 3 ⊢ (∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦) → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = 𝑦)
10	8, 9	syl 17	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = 𝑦)
11	1, 10	eqtrid 2782	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  {cab 2713  Vcvv 3459  {csn 4601  ∪ cuni 4883  ℩cio 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-sn 4602  df-pr 4604  df-uni 4884  df-iota 6484

This theorem is referenced by:  iotauni2  6500  iotaval  6502  iotaex  6504