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Theorem euxfr2dc 2778
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
euxfr2dc.1 𝐴 ∈ V
euxfr2dc.2 ∃*𝑦 𝑥 = 𝐴
Assertion
Ref Expression
euxfr2dc (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem euxfr2dc
StepHypRef Expression
1 euxfr2dc.2 . . . . . . 7 ∃*𝑦 𝑥 = 𝐴
21moani 2012 . . . . . 6 ∃*𝑦(𝜑𝑥 = 𝐴)
3 ancom 262 . . . . . . 7 ((𝜑𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜑))
43mobii 1979 . . . . . 6 (∃*𝑦(𝜑𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴𝜑))
52, 4mpbi 143 . . . . 5 ∃*𝑦(𝑥 = 𝐴𝜑)
65ax-gen 1379 . . . 4 𝑥∃*𝑦(𝑥 = 𝐴𝜑)
7 excom 1595 . . . . . 6 (∃𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥𝑦(𝑥 = 𝐴𝜑))
87dcbii 781 . . . . 5 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) ↔ DECID𝑥𝑦(𝑥 = 𝐴𝜑))
9 2euswapdc 2033 . . . . 5 (DECID𝑥𝑦(𝑥 = 𝐴𝜑) → (∀𝑥∃*𝑦(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑))))
108, 9sylbi 119 . . . 4 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∀𝑥∃*𝑦(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑))))
116, 10mpi 15 . . 3 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑)))
12 moeq 2768 . . . . . . 7 ∃*𝑥 𝑥 = 𝐴
1312moani 2012 . . . . . 6 ∃*𝑥(𝜑𝑥 = 𝐴)
143mobii 1979 . . . . . 6 (∃*𝑥(𝜑𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴𝜑))
1513, 14mpbi 143 . . . . 5 ∃*𝑥(𝑥 = 𝐴𝜑)
1615ax-gen 1379 . . . 4 𝑦∃*𝑥(𝑥 = 𝐴𝜑)
17 2euswapdc 2033 . . . 4 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∀𝑦∃*𝑥(𝑥 = 𝐴𝜑) → (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑))))
1816, 17mpi 15 . . 3 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑦𝑥(𝑥 = 𝐴𝜑) → ∃!𝑥𝑦(𝑥 = 𝐴𝜑)))
1911, 18impbid 127 . 2 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝑥(𝑥 = 𝐴𝜑)))
20 euxfr2dc.1 . . . 4 𝐴 ∈ V
21 biidd 170 . . . 4 (𝑥 = 𝐴 → (𝜑𝜑))
2220, 21ceqsexv 2639 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑)
2322eubii 1951 . 2 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
2419, 23syl6bb 194 1 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  DECID wdc 776  wal 1283   = wceq 1285  wex 1422  wcel 1434  ∃!weu 1942  ∃*wmo 1943  Vcvv 2602
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by:  euxfrdc  2779
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