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Theorem cbvreuvw 3361
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3365 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
cbvralvw.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 2820 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
2 cbvralvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2anbi12d 634 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
43cbveuvw 2605 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
5 df-reu 3068 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
6 df-reu 3068 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
74, 5, 63bitr4i 306 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2110  ∃!weu 2567  ∃!wreu 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-mo 2539  df-eu 2568  df-clel 2816  df-reu 3068
This theorem is referenced by:  reu8  3646  aceq1  9731  aceq2  9733  fin23lem27  9942  divalglem10  15963  lspsneu  20160  lshpsmreu  36860  wessf1ornlem  42395  fourierdlem50  43372
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