Theorem riota5 5513

Description: A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)

Hypotheses

Ref	Expression
riota5.1	⊢ (𝜑 → 𝐵 ∈ 𝐴)
riota5.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))

Assertion

Ref	Expression
riota5	⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem riota5

Step	Hyp	Ref	Expression
1		nfcvd 2220	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
2		riota5.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		riota5.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))
4	1, 2, 3	riota5f 5512	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ℩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-ral 2353  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:  f1ocnvfv3  5521  caucvgrelemrec  9865  sqrt0  9890  sqrtsq  9930  dfgcd3  10399