Theorem riota5 7294

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 2906	. 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 7293	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ℩crio 7263

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-reu 3305  df-v 3439  df-sbc 3722  df-un 3897  df-in 3899  df-ss 3909  df-sn 4566  df-pr 4568  df-uni 4845  df-iota 6410  df-riota 7264

This theorem is referenced by:  f1ocnvfv3  7303  ttrcltr  9522  sqrt0  15002  lubid  18129  lubun  18282  odval2  19208  adjvalval  30348  xdivpnfrp  31256  xrsinvgval  31335  dfgcd3  35543  poimirlem6  35831  poimirlem7  35832  lub0N  37403  glb0N  37407  trlval2  38377  cdlemefrs32fva  38614  cdleme32fva  38651  cdlemg1a  38784  unxpwdom3  41116