Theorem iotaval2 6463

Description: Version of iotaval 6466 using df-iota 6448 instead of dfiota2 6449. (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 6448	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
2		eqeq1 2744	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤}))
3		sneqbg 4781	. . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤))
4	3	elv 3437	. . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)
5		equcom 2025	. . . . . 6 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
6	4, 5	bitri 276	. . . . 5 ⊢ ({𝑦} = {𝑤} ↔ 𝑤 = 𝑦)
7	2, 6	bitrdi 288	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
8	7	alrimiv 1934	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦))
9		uniabio 6462	. . 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 2787	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545   = wceq 1547  {cab 2718  Vcvv 3432  {csn 4562  ∪ cuni 4845  ℩cio 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-sn 4563  df-pr 4565  df-uni 4846  df-iota 6448

This theorem is referenced by:  iotauni2  6464  iotaval  6466  iotaex  6468