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Theorem reuxfr 3740
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 3739 . 2 (⊤ → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑))
65mptru 1540 1 (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wtru 1534  wcel 2098  wrex 3064  ∃!wreu 3368  ∃*wrmo 3369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371
This theorem is referenced by: (None)
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