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

Theorem cbvrmow 3391
 Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3396 with a disjoint variable condition, which does not require ax-10 2142, ax-13 2379. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 23-May-2024.)
Hypotheses
Ref Expression
cbvrmow.1 𝑦𝜑
cbvrmow.2 𝑥𝜓
cbvrmow.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmow (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvrmow
StepHypRef Expression
1 nfv 1915 . . . 4 𝑦 𝑥𝐴
2 cbvrmow.1 . . . 4 𝑦𝜑
31, 2nfan 1900 . . 3 𝑦(𝑥𝐴𝜑)
4 nfv 1915 . . . 4 𝑥 𝑦𝐴
5 cbvrmow.2 . . . 4 𝑥𝜓
64, 5nfan 1900 . . 3 𝑥(𝑦𝐴𝜓)
7 eleq1w 2872 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
8 cbvrmow.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
97, 8anbi12d 633 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
103, 6, 9cbvmow 2663 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ ∃*𝑦(𝑦𝐴𝜓))
11 df-rmo 3114 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
12 df-rmo 3114 . 2 (∃*𝑦𝐴 𝜓 ↔ ∃*𝑦(𝑦𝐴𝜓))
1310, 11, 123bitr4i 306 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2111  ∃*wmo 2596  ∃*wrmo 3109 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-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-mo 2598  df-clel 2870  df-rmo 3114 This theorem is referenced by:  cbvdisj  5009
 Copyright terms: Public domain W3C validator