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Theorem euxfrdc 2721
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 (DECID yx(x = A ψ) → (∃!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 1927 . . . . . 6 (∃!y x = Ay x = A)
31, 2ax-mp 7 . . . . 5 y x = A
43biantrur 287 . . . 4 (φ ↔ (y x = A φ))
5 19.41v 1779 . . . 4 (y(x = A φ) ↔ (y x = A φ))
6 euxfrdc.3 . . . . . 6 (x = A → (φψ))
76pm5.32i 427 . . . . 5 ((x = A φ) ↔ (x = A ψ))
87exbii 1493 . . . 4 (y(x = A φ) ↔ y(x = A ψ))
94, 5, 83bitr2i 197 . . 3 (φy(x = A ψ))
109eubii 1906 . 2 (∃!xφ∃!xy(x = A ψ))
11 euxfrdc.1 . . 3 A V
121eumoi 1930 . . 3 ∃*y x = A
1311, 12euxfr2dc 2720 . 2 (DECID yx(x = A ψ) → (∃!xy(x = A ψ) ↔ ∃!yψ))
1410, 13syl5bb 181 1 (DECID yx(x = A ψ) → (∃!xφ∃!yψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  DECID wdc 741   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  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-bndl 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: (None)
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