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

Theorem cbvreuw 3408
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3425 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2139. (Revised by Wolf Lammen, 10-Dec-2024.)
Hypotheses
Ref Expression
cbvreuw.1 𝑦𝜑
cbvreuw.2 𝑥𝜓
cbvreuw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuw (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvreuw
StepHypRef Expression
1 cbvreuw.1 . . . 4 𝑦𝜑
2 cbvreuw.2 . . . 4 𝑥𝜓
3 cbvreuw.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrexw 3305 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
51, 2, 3cbvrmow 3407 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
64, 5anbi12i 628 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑) ↔ (∃𝑦𝐴 𝜓 ∧ ∃*𝑦𝐴 𝜓))
7 reu5 3380 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑))
8 reu5 3380 . 2 (∃!𝑦𝐴 𝜓 ↔ (∃𝑦𝐴 𝜓 ∧ ∃*𝑦𝐴 𝜓))
96, 7, 83bitr4i 303 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wnf 1780  wrex 3068  ∃!wreu 3376  ∃*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-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  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:  reu8nf  3886  poimirlem25  37632
  Copyright terms: Public domain W3C validator