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Theorem euxfrdc 2873
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
euxfrdc.1 𝐴 ∈ V
euxfrdc.2 ∃!𝑦 𝑥 = 𝐴
euxfrdc.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
euxfrdc (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem euxfrdc
StepHypRef Expression
1 euxfrdc.2 . . . . . 6 ∃!𝑦 𝑥 = 𝐴
2 euex 2030 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 5 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 301 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1875 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrdc.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 450 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1585 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 207 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 2009 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrdc.1 . . 3 𝐴 ∈ V
121eumoi 2033 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2dc 2872 . 2 (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓))
1410, 13syl5bb 191 1 (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  DECID wdc 820   = wceq 1332  wex 1469  wcel 1481  ∃!weu 2000  Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691
This theorem is referenced by: (None)
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