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Theorem reuxfr1 3730
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5309 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
reuxfr1.1 (𝑦𝐶𝐴𝐵)
reuxfr1.2 (𝑥𝐵 → ∃!𝑦𝐶 𝑥 = 𝐴)
reuxfr1.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
reuxfr1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐶 𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem reuxfr1
StepHypRef Expression
1 reuxfr1.1 . . . 4 (𝑦𝐶𝐴𝐵)
21adantl 485 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 reuxfr1.2 . . . 4 (𝑥𝐵 → ∃!𝑦𝐶 𝑥 = 𝐴)
43adantl 485 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
5 reuxfr1.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
62, 4, 5reuxfr1ds 3729 . 2 (⊤ → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐶 𝜓))
76mptru 1545 1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐶 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wtru 1539  wcel 2115  ∃!wreu 3135
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-mo 2624  df-eu 2655  df-cleq 2817  df-clel 2896  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141
This theorem is referenced by:  zmax  12345  rebtwnz  12347
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