Theorem riotaprop 7371

Description: Properties of a restricted definite description operator. (Contributed by NM, 23-Nov-2013.)

Hypotheses

Ref	Expression
riotaprop.0	⊢ Ⅎ𝑥𝜓
riotaprop.1	⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑)
riotaprop.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
riotaprop	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem riotaprop

Step	Hyp	Ref	Expression
1		riotaprop.1	. . 3 ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑)
2		riotacl 7361	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
3	1, 2	eqeltrid 2832	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴)
4	1	eqcomi 2738	. . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵
5		nfriota1 7351	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
6	1, 5	nfcxfr 2889	. . . . 5 ⊢ Ⅎ𝑥𝐵
7		riotaprop.0	. . . . 5 ⊢ Ⅎ𝑥𝜓
8		riotaprop.2	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
9	6, 7, 8	riota2f 7368	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
10	4, 9	mpbiri 258	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → 𝜓)
11	3, 10	mpancom 688	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓)
12	3, 11	jca 511	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∃!wreu 3352  ℩crio 7343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-un 3919  df-ss 3931  df-sn 4590  df-pr 4592  df-uni 4872  df-iota 6464  df-riota 7344

This theorem is referenced by:  fin23lem27  10281  lble  12135  ltrniotaval  40575