Theorem iotavalOLD 6470

Description: Obsolete version of iotaval 6467 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
iotavalOLD	⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotavalOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 6449	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
2		sbeqalb 3807	. . . . . . . 8 ⊢ (𝑦 ∈ V → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧))
3	2	elv 3451	. . . . . . 7 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧)
4	3	ex 413	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑦 = 𝑧))
5		equequ2 2029	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
6	5	bibi2d 342	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
7	6	biimpd 228	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 ↔ 𝑥 = 𝑧)))
8	7	alimdv 1919	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
9	8	com12 32	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑦 = 𝑧 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
10	4, 9	impbid 211	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑦 = 𝑧))
11		equcom 2021	. . . . 5 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
12	10, 11	bitrdi 286	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
13	12	alrimiv 1930	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
14		uniabio 6463	. . 3 ⊢ (∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = 𝑦)
15	13, 14	syl 17	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = 𝑦)
16	1, 15	eqtrid 2788	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541  {cab 2713  Vcvv 3445  ∪ cuni 4865  ℩cio 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-un 3915  df-in 3917  df-ss 3927  df-sn 4587  df-pr 4589  df-uni 4866  df-iota 6448

This theorem is referenced by: (None)