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Theorem reuxfr 3744
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 481 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃*𝑦𝐶 𝑥 = 𝐴)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
52, 4reuxfrd 3743 . 2 (⊤ → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑))
65mptru 1541 1 (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wtru 1535  wcel 2099  wrex 3067  ∃!wreu 3371  ∃*wrmo 3372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374
This theorem is referenced by: (None)
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