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Theorem reuxfr 4820
 Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 4822 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 481 . . 3 ((⊤ ∧ 𝑦𝐵) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)
5 reuxfr.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
62, 4, 5reuxfrd 4819 . 2 (⊤ → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓))
76trud 1484 1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∃!wreu 2898 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-v 3175 This theorem is referenced by:  zmax  11661  rebtwnz  11663
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