Theorem riotaprop 7415

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 7405	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
3	1, 2	eqeltrid 2843	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴)
4	1	eqcomi 2744	. . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵
5		nfriota1 7395	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
6	1, 5	nfcxfr 2901	. . . . 5 ⊢ Ⅎ𝑥𝐵
7		riotaprop.0	. . . . 5 ⊢ Ⅎ𝑥𝜓
8		riotaprop.2	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
9	6, 7, 8	riota2f 7412	. . . 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 1537  Ⅎwnf 1780   ∈ wcel 2106  ∃!wreu 3376  ℩crio 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-riota 7388

This theorem is referenced by:  fin23lem27  10366  lble  12218  ltrniotaval  40564