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Theorem reuxfr 4854
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 4856 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 482 . . 3 ((⊤ ∧ 𝑦𝐵) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)
43adantl 482 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)
5 reuxfr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
62, 4, 5reuxfrd 4853 . 2 (⊤ → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓))
76trud 1490 1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wtru 1481  wcel 1987  ∃!wreu 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-v 3188
This theorem is referenced by:  zmax  11729  rebtwnz  11731
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