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Theorem euxfrdc 2839
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 2005 . . . . . 6 (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
31, 2ax-mp 7 . . . . 5 𝑦 𝑥 = 𝐴
43biantrur 299 . . . 4 (𝜑 ↔ (∃𝑦 𝑥 = 𝐴𝜑))
5 19.41v 1856 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ (∃𝑦 𝑥 = 𝐴𝜑))
6 euxfrdc.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
76pm5.32i 447 . . . . 5 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
87exbii 1567 . . . 4 (∃𝑦(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴𝜓))
94, 5, 83bitr2i 207 . . 3 (𝜑 ↔ ∃𝑦(𝑥 = 𝐴𝜓))
109eubii 1984 . 2 (∃!𝑥𝜑 ↔ ∃!𝑥𝑦(𝑥 = 𝐴𝜓))
11 euxfrdc.1 . . 3 𝐴 ∈ V
121eumoi 2008 . . 3 ∃*𝑦 𝑥 = 𝐴
1311, 12euxfr2dc 2838 . 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 802   = wceq 1314  wex 1451  wcel 1463  ∃!weu 1975  Vcvv 2657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2659
This theorem is referenced by: (None)
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