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Theorem riotarab 7431
Description: Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
Hypothesis
Ref Expression
riotarab.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
riotarab (𝑥 ∈ {𝑦𝐴𝜓}𝜒) = (𝑥𝐴 (𝜑𝜒))
Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem riotarab
StepHypRef Expression
1 riotarab.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
21bicomd 223 . . . . . . 7 (𝑥 = 𝑦 → (𝜓𝜑))
32equcoms 2018 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
43elrab 3691 . . . . 5 (𝑥 ∈ {𝑦𝐴𝜓} ↔ (𝑥𝐴𝜑))
54anbi1i 624 . . . 4 ((𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒) ↔ ((𝑥𝐴𝜑) ∧ 𝜒))
6 anass 468 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜑𝜒)))
75, 6bitri 275 . . 3 ((𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜑𝜒)))
87iotabii 6545 . 2 (℩𝑥(𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒)) = (℩𝑥(𝑥𝐴 ∧ (𝜑𝜒)))
9 df-riota 7389 . 2 (𝑥 ∈ {𝑦𝐴𝜓}𝜒) = (℩𝑥(𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒))
10 df-riota 7389 . 2 (𝑥𝐴 (𝜑𝜒)) = (℩𝑥(𝑥𝐴 ∧ (𝜑𝜒)))
118, 9, 103eqtr4i 2774 1 (𝑥 ∈ {𝑦𝐴𝜓}𝜒) = (𝑥𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {crab 3435  cio 6511  crio 7388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-ss 3967  df-uni 4907  df-iota 6513  df-riota 7389
This theorem is referenced by:  eqscut  27851
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