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Theorem rmoeq1 3406
Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 30-Apr-2023.)
Assertion
Ref Expression
rmoeq1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoeq1
StepHypRef Expression
1 eleq2 2898 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 629 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
32mobidv 2626 . 2 (𝐴 = 𝐵 → (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐵𝜑)))
4 df-rmo 3143 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
5 df-rmo 3143 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
63, 4, 53bitr4g 315 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ∃*wmo 2613  ∃*wrmo 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-mo 2615  df-cleq 2811  df-clel 2890  df-rmo 3143
This theorem is referenced by:  rmoeqd  3418  rmosn  4647  poimirlem2  34775
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