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

Theorem reuxfr 5091
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5093 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
reuxfr.1 (𝑦𝐵𝐴𝐵)
reuxfr.2 (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)
reuxfr.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reuxfr (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr
StepHypRef Expression
1 reuxfr.1 . . . 4 (𝑦𝐵𝐴𝐵)
21adantl 474 . . 3 ((⊤ ∧ 𝑦𝐵) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)
43adantl 474 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)
5 reuxfr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
62, 4, 5reuxfrd 5090 . 2 (⊤ → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓))
76mptru 1661 1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wtru 1654  wcel 2157  ∃!wreu 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-v 3386
This theorem is referenced by:  zmax  12027  rebtwnz  12029
  Copyright terms: Public domain W3C validator