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Theorem cbvreuvw 3376
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3401 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.)
Hypothesis
Ref Expression
cbvrmovw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuvw (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuvw
StepHypRef Expression
1 eleq1w 2821 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvrmovw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 632 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43cbveuvw 2606 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
5 df-reu 3353 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
6 df-reu 3353 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
74, 5, 63bitr4i 303 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  ∃!weu 2568  ∃!wreu 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-mo 2540  df-eu 2569  df-clel 2816  df-reu 3353
This theorem is referenced by:  reu8  3690  aceq1  9987  aceq2  9989  fin23lem27  10198  divalglem10  16220  lspsneu  20513  lshpsmreu  37502  wessf1ornlem  43198  fourierdlem50  44188
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