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Theorem reuxfr1 3761
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5426 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 481 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 reuxfr1.2 . . . 4 (𝑥𝐵 → ∃!𝑦𝐶 𝑥 = 𝐴)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)
5 reuxfr1.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
62, 4, 5reuxfr1ds 3760 . 2 (⊤ → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐶 𝜓))
76mptru 1544 1 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐶 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wtru 1538  wcel 2106  ∃!wreu 3376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379
This theorem is referenced by:  zmax  12985  rebtwnz  12987
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