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Theorem euxfr2dc 2869
 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 2069 . . . . . 6 ∃*𝑦(𝜑𝑥 = 𝐴)
3 ancom 264 . . . . . . 7 ((𝜑𝑥 = 𝐴) ↔ (𝑥 = 𝐴𝜑))
43mobii 2036 . . . . . 6 (∃*𝑦(𝜑𝑥 = 𝐴) ↔ ∃*𝑦(𝑥 = 𝐴𝜑))
52, 4mpbi 144 . . . . 5 ∃*𝑦(𝑥 = 𝐴𝜑)
65ax-gen 1425 . . . 4 𝑥∃*𝑦(𝑥 = 𝐴𝜑)
7 excom 1642 . . . . . 6 (∃𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥𝑦(𝑥 = 𝐴𝜑))
87dcbii 825 . . . . 5 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) ↔ DECID𝑥𝑦(𝑥 = 𝐴𝜑))
9 2euswapdc 2090 . . . . 5 (DECID𝑥𝑦(𝑥 = 𝐴𝜑) → (∀𝑥∃*𝑦(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) → ∃!𝑦𝑥(𝑥 = 𝐴𝜑))))
108, 9sylbi 120 . . . 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 2859 . . . . . . 7 ∃*𝑥 𝑥 = 𝐴
1312moani 2069 . . . . . 6 ∃*𝑥(𝜑𝑥 = 𝐴)
143mobii 2036 . . . . . 6 (∃*𝑥(𝜑𝑥 = 𝐴) ↔ ∃*𝑥(𝑥 = 𝐴𝜑))
1513, 14mpbi 144 . . . . 5 ∃*𝑥(𝑥 = 𝐴𝜑)
1615ax-gen 1425 . . . 4 𝑦∃*𝑥(𝑥 = 𝐴𝜑)
17 2euswapdc 2090 . . . 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 128 . 2 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝑥(𝑥 = 𝐴𝜑)))
20 euxfr2dc.1 . . . 4 𝐴 ∈ V
21 biidd 171 . . . 4 (𝑥 = 𝐴 → (𝜑𝜑))
2220, 21ceqsexv 2725 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑)
2322eubii 2008 . 2 (∃!𝑦𝑥(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
2419, 23syl6bb 195 1 (DECID𝑦𝑥(𝑥 = 𝐴𝜑) → (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃!weu 1999  ∃*wmo 2000  Vcvv 2686 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688 This theorem is referenced by:  euxfrdc  2870
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