Theorem iotaval2 6488

Description: Version of iotaval 6491 using df-iota 6473 instead of dfiota2 6474. (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 6473	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		eqeq1 2765	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤}))
3		sneqbg 4800	. . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤))
4	3	elv 3458	. . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)
5		equcom 2037	. . . . . 6 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
6	4, 5	bitri 277	. . . . 5 ⊢ ({𝑦} = {𝑤} ↔ 𝑤 = 𝑦)
7	2, 6	bitrdi 289	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
8	7	alrimiv 1946	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
9		uniabio 6487	. . 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 2808	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557   = wceq 1559  {cab 2739  Vcvv 3453  {csn 4581  ∪ cuni 4864  ℩cio 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-ss 3921  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6473

This theorem is referenced by:  iotauni2  6489  iotaval  6491  iotaex  6493