Theorem riota5 7353

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 2899	. 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 7352	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ℩crio 7323

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 2708

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-reu 3343  df-v 3431  df-sbc 3729  df-un 3894  df-ss 3906  df-sn 4568  df-pr 4570  df-uni 4851  df-iota 6454  df-riota 7324

This theorem is referenced by:  f1ocnvfv3  7362  ttrcltr  9637  sqrt0  15203  lubid  18326  lubun  18481  odval2  19526  adjvalval  32008  xdivpnfrp  32992  xrsinvgval  33068  dfgcd3  37638  poimirlem6  37947  poimirlem7  37948  lub0N  39635  glb0N  39639  trlval2  40609  cdlemefrs32fva  40846  cdleme32fva  40883  cdlemg1a  41016  unxpwdom3  43523