Theorem riota2f 5759

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 2293	. 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 275	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓))
9	2, 3, 5, 6, 8	riota2df 5758	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  Ⅎwnf 1437   ∈ wcel 1481  Ⅎwnfc 2269  ∃!wreu 2419  ℩crio 5737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-reu 2424  df-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096  df-riota 5738

This theorem is referenced by:  riota2  5760  riotaprop  5761  riotass2  5764  riotass  5765