Theorem riota2f 5509

Description: This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota2f.1	⊢ Ⅎ𝑥𝐵
riota2f.2	⊢ Ⅎ𝑥𝜓
riota2f.3	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
riota2f	⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem riota2f

Step	Hyp	Ref	Expression
1		riota2f.1	. . 3 ⊢ Ⅎ𝑥𝐵
2	1	nfel1 2229	. 2 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
3	1	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝐵)
4		riota2f.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	4	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝜓)
6		id 19	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴)
7		riota2f.3	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
8	7	adantl 271	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓))
9	2, 3, 5, 6, 8	riota2df 5508	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  Ⅎwnfc 2206  ∃!wreu 2350  ℩crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  riota2  5510  riotaprop  5511  riotass2  5514  riotass  5515