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Theorem euxfr2 3021
 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
euxfr2.1 A V
euxfr2.2 ∃*y x = A
Assertion
Ref Expression
euxfr2 (∃!xy(x = A φ) ↔ ∃!yφ)
Distinct variable groups:   φ,x   x,A
Allowed substitution hints:   φ(y)   A(y)

Proof of Theorem euxfr2
StepHypRef Expression
1 2euswap 2280 . . . 4 (x∃*y(x = A φ) → (∃!xy(x = A φ) → ∃!yx(x = A φ)))
2 euxfr2.2 . . . . . 6 ∃*y x = A
32moani 2256 . . . . 5 ∃*y(φ x = A)
4 ancom 437 . . . . . 6 ((φ x = A) ↔ (x = A φ))
54mobii 2240 . . . . 5 (∃*y(φ x = A) ↔ ∃*y(x = A φ))
63, 5mpbi 199 . . . 4 ∃*y(x = A φ)
71, 6mpg 1548 . . 3 (∃!xy(x = A φ) → ∃!yx(x = A φ))
8 2euswap 2280 . . . 4 (y∃*x(x = A φ) → (∃!yx(x = A φ) → ∃!xy(x = A φ)))
9 moeq 3012 . . . . . 6 ∃*x x = A
109moani 2256 . . . . 5 ∃*x(φ x = A)
114mobii 2240 . . . . 5 (∃*x(φ x = A) ↔ ∃*x(x = A φ))
1210, 11mpbi 199 . . . 4 ∃*x(x = A φ)
138, 12mpg 1548 . . 3 (∃!yx(x = A φ) → ∃!xy(x = A φ))
147, 13impbii 180 . 2 (∃!xy(x = A φ) ↔ ∃!yx(x = A φ))
15 euxfr2.1 . . . 4 A V
16 biidd 228 . . . 4 (x = A → (φφ))
1715, 16ceqsexv 2894 . . 3 (x(x = A φ) ↔ φ)
1817eubii 2213 . 2 (∃!yx(x = A φ) ↔ ∃!yφ)
1914, 18bitri 240 1 (∃!xy(x = A φ) ↔ ∃!yφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  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:  euxfr  3022
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