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

Theorem cbvreuv 3459
 Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3456 for a version without ax-13 2385, but extra disjoint variables. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
cbvralv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuv (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuv
StepHypRef Expression
1 nfv 1908 . 2 𝑦𝜑
2 nfv 1908 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvreu 3452 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∃!wreu 3144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clel 2897  df-reu 3149 This theorem is referenced by:  lshpsmreu  36112  wessf1ornlem  41306  fourierdlem50  42303
 Copyright terms: Public domain W3C validator