Theorem riotaprop 5645

Description: Properties of a restricted definite description operator. Todo (df-riota 5622 update): can some uses of riota2f 5643 be shortened with this? (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 5636	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
3	1, 2	syl5eqel 2175	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴)
4	1	eqcomi 2093	. . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵
5		nfriota1 5629	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
6	1, 5	nfcxfr 2226	. . . . 5 ⊢ Ⅎ𝑥𝐵
7		riotaprop.0	. . . . 5 ⊢ Ⅎ𝑥𝜓
8		riotaprop.2	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
9	6, 7, 8	riota2f 5643	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
10	4, 9	mpbiri 167	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → 𝜓)
11	3, 10	mpancom 414	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓)
12	3, 11	jca 301	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290  Ⅎwnf 1395   ∈ wcel 1439  ∃!wreu 2362  ℩crio 5621

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-uni 3660  df-iota 4993  df-riota 5622

This theorem is referenced by:  lble  8469