Theorem iotaval2 6502

Description: Version of iotaval 6505 using df-iota 6487 instead of dfiota2 6488. (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 6487	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		eqeq1 2765	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤}))
3		sneqbg 4803	. . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤))
4	3	elv 3456	. . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)
5		equcom 2051	. . . . . 6 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
6	4, 5	bitri 278	. . . . 5 ⊢ ({𝑦} = {𝑤} ↔ 𝑤 = 𝑦)
7	2, 6	bitrdi 290	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
8	7	alrimiv 1960	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
9		uniabio 6501	. . 3 ⊢ (∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦) → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = 𝑦)
10	8, 9	syl 18	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = 𝑦)
11	1, 10	eqtrid 2808	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  {cab 2739  Vcvv 3451  {csn 4584  ∪ cuni 4867  ℩cio 6485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6487

This theorem is used by:  iotauni2  6503  iotaval  6505  iotaex  6507