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

Theorem cbvrexv 3400
 Description: Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw 3397 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrexvw 3397 when possible. (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbvralv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexv (∃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 cbvrexv
StepHypRef Expression
1 nfv 1915 . 2 𝑦𝜑
2 nfv 1915 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrex 3393 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∃wrex 3107 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  ax-13 2379 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-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112 This theorem is referenced by:  cbvrex2v  3412  cygablOLD  19007  rexlimdvaacbv  40926
 Copyright terms: Public domain W3C validator