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Theorem euxfrdc 2937
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 2067 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 5 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 303 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1913 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrdc.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 454 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1615 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 208 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 2046 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrdc.1 . . 3 𝐴 ∈ V
121eumoi 2070 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2dc 2936 . 2 (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝑦(𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝜓))
1410, 13bitrid 192 1 (DECID𝑦𝑥(𝑥 = 𝐴𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 835   = wceq 1363  wex 1502  ∃!weu 2037  wcel 2159  Vcvv 2751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-v 2753
This theorem is referenced by: (None)
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