Theorem riota5 7335

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 2892	. 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 7334	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ℩crio 7305

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-reu 3344  df-v 3438  df-sbc 3743  df-un 3908  df-ss 3920  df-sn 4578  df-pr 4580  df-uni 4859  df-iota 6438  df-riota 7306

This theorem is referenced by:  f1ocnvfv3  7344  ttrcltr  9612  sqrt0  15148  lubid  18266  lubun  18421  odval2  19430  adjvalval  31885  xdivpnfrp  32882  xrsinvgval  32971  dfgcd3  37318  poimirlem6  37626  poimirlem7  37627  lub0N  39188  glb0N  39192  trlval2  40162  cdlemefrs32fva  40399  cdleme32fva  40436  cdlemg1a  40569  unxpwdom3  43088