Theorem riotaprop 5979

Description: Properties of a restricted definite description operator. Todo (df-riota 5953 update): can some uses of riota2f 5976 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 5969	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴)
3	1, 2	eqeltrid 2316	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴)
4	1	eqcomi 2233	. . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵
5		nfriota1 5961	. . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑)
6	1, 5	nfcxfr 2369	. . . . 5 ⊢ Ⅎ𝑥𝐵
7		riotaprop.0	. . . . 5 ⊢ Ⅎ𝑥𝜓
8		riotaprop.2	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
9	6, 7, 8	riota2f 5976	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))
10	4, 9	mpbiri 168	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → 𝜓)
11	3, 10	mpancom 422	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓)
12	3, 11	jca 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  Ⅎwnf 1506   ∈ wcel 2200  ∃!wreu 2510  ℩crio 5952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-uni 3888  df-iota 5277  df-riota 5953

This theorem is referenced by:  lble  9090