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Theorem reuxfr 3737
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 482 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 reuxfr.2 . . . 4 (𝑥𝐵 → ∃*𝑦𝐶 𝑥 = 𝐴)
43adantl 482 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)
52, 4reuxfrd 3736 . 2 (⊤ → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑))
65mptru 1535 1 (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐶 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wtru 1529  wcel 2105  wrex 3136  ∃!wreu 3137  ∃*wrmo 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143
This theorem is referenced by: (None)
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