Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvreuvw Structured version   Visualization version   GIF version

Theorem cbvreuvw 3399
 Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3402 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-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 nfv 1915 . 2 𝑦𝜑
2 nfv 1915 . 2 𝑥𝜓
3 cbvralvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvreuw 3390 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∃!wreu 3108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clel 2870  df-reu 3113 This theorem is referenced by:  reu8  3674  aceq1  9546  aceq2  9548  fin23lem27  9757  divalglem10  15763  lspsneu  19909  lshpsmreu  36556  wessf1ornlem  41981  fourierdlem50  42966
 Copyright terms: Public domain W3C validator