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Theorem reuxfr2 4922
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
reuxfr2.1 (𝑦𝐵𝐴𝐵)
reuxfr2.2 (𝑥𝐵 → ∃*𝑦𝐵 𝑥 = 𝐴)
Assertion
Ref Expression
reuxfr2 (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐵 𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr2
StepHypRef Expression
1 reuxfr2.1 . . . 4 (𝑦𝐵𝐴𝐵)
21adantl 481 . . 3 ((⊤ ∧ 𝑦𝐵) → 𝐴𝐵)
3 reuxfr2.2 . . . 4 (𝑥𝐵 → ∃*𝑦𝐵 𝑥 = 𝐴)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)
52, 4reuxfr2d 4921 . 2 (⊤ → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐵 𝜑))
65trud 1533 1 (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wtru 1524  wcel 2030  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-v 3233
This theorem is referenced by: (None)
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