Theorem iotaval2 6510

Description: Version of iotaval 6513 using df-iota 6494 instead of dfiota2 6495. (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 6494	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		eqeq1 2731	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤}))
3		sneqbg 4840	. . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤))
4	3	elv 3475	. . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)
5		equcom 2014	. . . . . 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 1923	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
9		uniabio 6509	. . 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 2779	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534  {cab 2704  Vcvv 3469  {csn 4624  ∪ cuni 4903  ℩cio 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-un 3949  df-in 3951  df-ss 3961  df-sn 4625  df-pr 4627  df-uni 4904  df-iota 6494

This theorem is referenced by:  iotauni2  6511  iotaval  6513  iotaex  6515