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Theorem cbvrmovw2 36171
Description: Change bound variable and domain in the restricted at-most-one quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvrmovw2.1 (𝑥 = 𝑦𝐴 = 𝐵)
cbvrmovw2.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmovw2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrmovw2
StepHypRef Expression
1 eleq1w 2820 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvrmovw2.1 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝐵)
32eleq2d 2823 . . . . 5 (𝑥 = 𝑦 → (𝑦𝐴𝑦𝐵))
41, 3bitrd 279 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
5 cbvrmovw2.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5anbi12d 631 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐵𝜓)))
76cbvmovw 2598 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑦(𝑦𝐵𝜓))
8 df-rmo 3376 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
9 df-rmo 3376 . 2 (∃*𝑦𝐵 𝜓 ↔ ∃*𝑦(𝑦𝐵𝜓))
107, 8, 93bitr4i 303 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  ∃*wmo 2534  ∃*wrmo 3375
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-ex 1775  df-mo 2536  df-cleq 2725  df-clel 2812  df-rmo 3376
This theorem is referenced by:  cbvdisjvw2  36178
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