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Theorem euxfrdc 2749
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 1946 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 7 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 291 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1798 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrdc.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 435 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1512 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 201 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 1925 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrdc.1 . . 3 𝐴 ∈ V
121eumoi 1949 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2dc 2748 . 2 (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓))
1410, 13syl5bb 185 1 (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  DECID wdc 753   = wceq 1259  wex 1397  wcel 1409  ∃!weu 1916  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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