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

Theorem reuxfr 3717
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
reuxfr.1 (𝑦𝐶𝐴𝐵)
reuxfr.2 (𝑥𝐵 → ∃*𝑦𝐶 𝑥 = 𝐴)
Assertion
Ref Expression
reuxfr (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr
StepHypRef Expression
1 reuxfr.1 . . . 4 (𝑦𝐶𝐴𝐵)
21adantl 485 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃*𝑦𝐶 𝑥 = 𝐴)
43adantl 485 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
52, 4reuxfrd 3716 . 2 (⊤ → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑))
65mptru 1545 1 (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wtru 1539  wcel 2115  wrex 3127  ∃!wreu 3128  ∃*wrmo 3129
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
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 2071  df-mo 2623  df-eu 2654  df-cleq 2814  df-clel 2892  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator