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Theorem reuxfr1ds 3693
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5288 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
reuxfr1ds.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
reuxfr1ds.2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
reuxfr1ds.3 (𝑥 = 𝐴 → (𝜓𝜒))
Assertion
Ref Expression
reuxfr1ds (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr1ds
StepHypRef Expression
1 reuxfr1ds.1 . 2 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 reuxfr1ds.2 . 2 ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
3 reuxfr1ds.3 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
43adantl 485 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
51, 2, 4reuxfr1d 3692 1 (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  ∃!wreu 3111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117
This theorem is referenced by:  reuxfr1  3694  riotaxfrd  7131
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