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Theorem rmoeq1 3414
Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2139, ax-11 2155, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2108. (Revised by Wolf Lammen, 12-Mar-2025.)
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
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2728 . . . . . 6 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
21biimpi 216 . . . . 5 (𝐴 = 𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
3 anbi1 633 . . . . . . 7 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
43imbi1d 341 . . . . . 6 ((𝑥𝐴𝑥𝐵) → (((𝑥𝐴𝜑) → 𝑥 = 𝑧) ↔ ((𝑥𝐵𝜑) → 𝑥 = 𝑧)))
54alimi 1808 . . . . 5 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥(((𝑥𝐴𝜑) → 𝑥 = 𝑧) ↔ ((𝑥𝐵𝜑) → 𝑥 = 𝑧)))
6 albi 1815 . . . . 5 (∀𝑥(((𝑥𝐴𝜑) → 𝑥 = 𝑧) ↔ ((𝑥𝐵𝜑) → 𝑥 = 𝑧)) → (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑧) ↔ ∀𝑥((𝑥𝐵𝜑) → 𝑥 = 𝑧)))
72, 5, 63syl 18 . . . 4 (𝐴 = 𝐵 → (∀𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑧) ↔ ∀𝑥((𝑥𝐵𝜑) → 𝑥 = 𝑧)))
87exbidv 1919 . . 3 (𝐴 = 𝐵 → (∃𝑧𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑧) ↔ ∃𝑧𝑥((𝑥𝐵𝜑) → 𝑥 = 𝑧)))
9 df-mo 2538 . . 3 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃𝑧𝑥((𝑥𝐴𝜑) → 𝑥 = 𝑧))
10 df-mo 2538 . . 3 (∃*𝑥(𝑥𝐵𝜑) ↔ ∃𝑧𝑥((𝑥𝐵𝜑) → 𝑥 = 𝑧))
118, 9, 103bitr4g 314 . 2 (𝐴 = 𝐵 → (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐵𝜑)))
12 df-rmo 3378 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
13 df-rmo 3378 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
1411, 12, 133bitr4g 314 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  ∃*wmo 2536  ∃*wrmo 3377
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-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-cleq 2727  df-rmo 3378
This theorem is referenced by:  reueq1  3415  rmoeqd  3418  rmosn  4724  rmoeqdv  36194  poimirlem2  37609
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