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

Theorem reuxfr 5043
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5045 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 473 . . 3 ((⊤ ∧ 𝑦𝐵) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)
43adantl 473 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)
5 reuxfr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
62, 4, 5reuxfrd 5042 . 2 (⊤ → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓))
76trud 1642 1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wtru 1633  wcel 2139  ∃!wreu 3052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-v 3342
This theorem is referenced by:  zmax  11998  rebtwnz  12000
  Copyright terms: Public domain W3C validator