Theorem riota2f 7341

Description: This theorem shows a condition that allows 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 2916	. 2 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
3	1	a1i 11	. 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝐵)
4		riota2f.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	4	a1i 11	. 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝜓)
6		id 22	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴)
7		riota2f.3	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
8	7	adantl 481	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓))
9	2, 3, 5, 6, 8	riota2df 7340	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884  ∃!wreu 3349  ℩crio 7316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-reu 3352  df-v 3443  df-un 3907  df-ss 3919  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6449  df-riota 7317

This theorem is referenced by:  riota2  7342  riotaprop  7344  riotass2  7347  riotass  7348  riotaxfrd  7351  ttrcltr  9629  cdlemksv2  41175  cdlemkuv2  41195  cdlemk36  41241