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Theorem cbvrmow 3409
Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3429 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2377. (Contributed by NM, 16-Jun-2017.) Avoid ax-10 2141, ax-13 2377. (Revised by GG, 23-May-2024.)
Hypotheses
Ref Expression
cbvrmow.1 𝑦𝜑
cbvrmow.2 𝑥𝜓
cbvrmow.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmow (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvrmow
StepHypRef Expression
1 nfv 1914 . . . 4 𝑦 𝑥𝐴
2 cbvrmow.1 . . . 4 𝑦𝜑
31, 2nfan 1899 . . 3 𝑦(𝑥𝐴𝜑)
4 nfv 1914 . . . 4 𝑥 𝑦𝐴
5 cbvrmow.2 . . . 4 𝑥𝜓
64, 5nfan 1899 . . 3 𝑥(𝑦𝐴𝜓)
7 eleq1w 2824 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
8 cbvrmow.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
97, 8anbi12d 632 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
103, 6, 9cbvmow 2603 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑦(𝑦𝐴𝜓))
11 df-rmo 3380 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
12 df-rmo 3380 . 2 (∃*𝑦𝐴 𝜓 ↔ ∃*𝑦(𝑦𝐴𝜓))
1310, 11, 123bitr4i 303 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wnf 1783  wcel 2108  ∃*wmo 2538  ∃*wrmo 3379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-clel 2816  df-rmo 3380
This theorem is referenced by:  cbvreuw  3410  cbvdisj  5120
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