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Theorem riotarab 7430
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 2017 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
43elrab 3695 . . . . 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 6548 . 2 (℩𝑥(𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒)) = (℩𝑥(𝑥𝐴 ∧ (𝜑𝜒)))
9 df-riota 7388 . 2 (𝑥 ∈ {𝑦𝐴𝜓}𝜒) = (℩𝑥(𝑥 ∈ {𝑦𝐴𝜓} ∧ 𝜒))
10 df-riota 7388 . 2 (𝑥𝐴 (𝜑𝜒)) = (℩𝑥(𝑥𝐴 ∧ (𝜑𝜒)))
118, 9, 103eqtr4i 2773 1 (𝑥 ∈ {𝑦𝐴𝜓}𝜒) = (𝑥𝐴 (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  cio 6514  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-ss 3980  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  eqscut  27865
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