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Theorem euxfr2dc 2720
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 (DECID yx(x = A φ) → (∃!xy(x = A φ) ↔ ∃!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 1967 . . . . . 6 ∃*y(φ x = A)
3 ancom 253 . . . . . . 7 ((φ x = A) ↔ (x = A φ))
43mobii 1934 . . . . . 6 (∃*y(φ x = A) ↔ ∃*y(x = A φ))
52, 4mpbi 133 . . . . 5 ∃*y(x = A φ)
65ax-gen 1335 . . . 4 x∃*y(x = A φ)
7 excom 1551 . . . . . 6 (yx(x = A φ) ↔ xy(x = A φ))
87dcbii 746 . . . . 5 (DECID yx(x = A φ) ↔ DECID xy(x = A φ))
9 2euswapdc 1988 . . . . 5 (DECID xy(x = A φ) → (x∃*y(x = A φ) → (∃!xy(x = A φ) → ∃!yx(x = A φ))))
108, 9sylbi 114 . . . 4 (DECID yx(x = A φ) → (x∃*y(x = A φ) → (∃!xy(x = A φ) → ∃!yx(x = A φ))))
116, 10mpi 15 . . 3 (DECID yx(x = A φ) → (∃!xy(x = A φ) → ∃!yx(x = A φ)))
12 moeq 2710 . . . . . . 7 ∃*x x = A
1312moani 1967 . . . . . 6 ∃*x(φ x = A)
143mobii 1934 . . . . . 6 (∃*x(φ x = A) ↔ ∃*x(x = A φ))
1513, 14mpbi 133 . . . . 5 ∃*x(x = A φ)
1615ax-gen 1335 . . . 4 y∃*x(x = A φ)
17 2euswapdc 1988 . . . 4 (DECID yx(x = A φ) → (y∃*x(x = A φ) → (∃!yx(x = A φ) → ∃!xy(x = A φ))))
1816, 17mpi 15 . . 3 (DECID yx(x = A φ) → (∃!yx(x = A φ) → ∃!xy(x = A φ)))
1911, 18impbid 120 . 2 (DECID yx(x = A φ) → (∃!xy(x = A φ) ↔ ∃!yx(x = A φ)))
20 euxfr2dc.1 . . . 4 A V
21 biidd 161 . . . 4 (x = A → (φφ))
2220, 21ceqsexv 2587 . . 3 (x(x = A φ) ↔ φ)
2322eubii 1906 . 2 (∃!yx(x = A φ) ↔ ∃!yφ)
2419, 23syl6bb 185 1 (DECID yx(x = A φ) → (∃!xy(x = A φ) ↔ ∃!yφ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  DECID wdc 741  wal 1240   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  ∃*wmo 1898  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  euxfrdc  2721
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