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Theorem riotaeqbii 36140
Description: Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
riotaeqbii.1 𝐴 = 𝐵
riotaeqbii.2 (𝜑𝜓)
Assertion
Ref Expression
riotaeqbii (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓)

Proof of Theorem riotaeqbii
StepHypRef Expression
1 riotaeqbii.1 . . . . 5 𝐴 = 𝐵
21eleq2i 2829 . . . 4 (𝑥𝐴𝑥𝐵)
3 riotaeqbii.2 . . . 4 (𝜑𝜓)
42, 3anbi12i 627 . . 3 ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))
54iotabii 6544 . 2 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑥(𝑥𝐵𝜓))
6 df-riota 7382 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
7 df-riota 7382 . 2 (𝑥𝐵 𝜓) = (℩𝑥(𝑥𝐵𝜓))
85, 6, 73eqtr4i 2771 1 (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1535  wcel 2104  cio 6509  crio 7381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479  df-ss 3980  df-uni 4916  df-iota 6511  df-riota 7382
This theorem is referenced by:  riotaeqi  36141
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