Theorem riota2f 7412

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 2922	. 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 7411	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890  ∃!wreu 3378  ℩crio 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-reu 3381  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-riota 7388

This theorem is referenced by:  riota2  7413  riotaprop  7415  riotass2  7418  riotass  7419  riotaxfrd  7422  ttrcltr  9756  cdlemksv2  40849  cdlemkuv2  40869  cdlemk36  40915