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Theorem euxfr 3022
 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
euxfr.1 A V
euxfr.2 ∃!y x = A
euxfr.3 (x = A → (φψ))
Assertion
Ref Expression
euxfr (∃!xφ∃!yψ)
Distinct variable groups:   ψ,x   φ,y   x,A
Allowed substitution hints:   φ(x)   ψ(y)   A(y)

Proof of Theorem euxfr
StepHypRef Expression
1 euxfr.2 . . . . . 6 ∃!y x = A
2 euex 2227 . . . . . 6 (∃!y x = Ay x = A)
31, 2ax-mp 5 . . . . 5 y x = A
43biantrur 492 . . . 4 (φ ↔ (y x = A φ))
5 19.41v 1901 . . . 4 (y(x = A φ) ↔ (y x = A φ))
6 euxfr.3 . . . . . 6 (x = A → (φψ))
76pm5.32i 618 . . . . 5 ((x = A φ) ↔ (x = A ψ))
87exbii 1582 . . . 4 (y(x = A φ) ↔ y(x = A ψ))
94, 5, 83bitr2i 264 . . 3 (φy(x = A ψ))
109eubii 2213 . 2 (∃!xφ∃!xy(x = A ψ))
11 euxfr.1 . . 3 A V
121eumoi 2245 . . 3 ∃*y x = A
1311, 12euxfr2 3021 . 2 (∃!xy(x = A ψ) ↔ ∃!yψ)
1410, 13bitri 240 1 (∃!xφ∃!yψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by: (None)
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