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

Theorem cbvrmov 3427
Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbvrmov.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmov (∃*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 cbvrmov
StepHypRef Expression
1 nfv 1912 . 2 𝑦𝜑
2 nfv 1912 . 2 𝑥𝜓
3 cbvrmov.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrmo 3426 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  ∃*wrmo 3377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator