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Theorem eu2 1941
Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eu2.1 yφ
Assertion
Ref Expression
eu2 (∃!xφ ↔ (xφ xy((φ [y / x]φ) → x = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem eu2
StepHypRef Expression
1 euex 1927 . . 3 (∃!xφxφ)
2 eu2.1 . . . . . 6 yφ
32nfri 1409 . . . . 5 (φyφ)
43eumo0 1928 . . . 4 (∃!xφyx(φx = y))
52mo23 1938 . . . 4 (yx(φx = y) → xy((φ [y / x]φ) → x = y))
64, 5syl 14 . . 3 (∃!xφxy((φ [y / x]φ) → x = y))
71, 6jca 290 . 2 (∃!xφ → (xφ xy((φ [y / x]φ) → x = y)))
8 19.29r 1509 . . . 4 ((xφ xy((φ [y / x]φ) → x = y)) → x(φ y((φ [y / x]φ) → x = y)))
9 impexp 250 . . . . . . . . 9 (((φ [y / x]φ) → x = y) ↔ (φ → ([y / x]φx = y)))
109albii 1356 . . . . . . . 8 (y((φ [y / x]φ) → x = y) ↔ y(φ → ([y / x]φx = y)))
11219.21 1472 . . . . . . . 8 (y(φ → ([y / x]φx = y)) ↔ (φy([y / x]φx = y)))
1210, 11bitri 173 . . . . . . 7 (y((φ [y / x]φ) → x = y) ↔ (φy([y / x]φx = y)))
1312anbi2i 430 . . . . . 6 ((φ y((φ [y / x]φ) → x = y)) ↔ (φ (φy([y / x]φx = y))))
14 abai 494 . . . . . 6 ((φ y([y / x]φx = y)) ↔ (φ (φy([y / x]φx = y))))
1513, 14bitr4i 176 . . . . 5 ((φ y((φ [y / x]φ) → x = y)) ↔ (φ y([y / x]φx = y)))
1615exbii 1493 . . . 4 (x(φ y((φ [y / x]φ) → x = y)) ↔ x(φ y([y / x]φx = y)))
178, 16sylib 127 . . 3 ((xφ xy((φ [y / x]φ) → x = y)) → x(φ y([y / x]φx = y)))
183eu1 1922 . . 3 (∃!xφx(φ y([y / x]φx = y)))
1917, 18sylibr 137 . 2 ((xφ xy((φ [y / x]φ) → x = y)) → ∃!xφ)
207, 19impbii 117 1 (∃!xφ ↔ (xφ xy((φ [y / x]φ) → x = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wnf 1346  wex 1378  [wsb 1642  ∃!weu 1897
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-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
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900
This theorem is referenced by:  eu3h  1942  mo3h  1950  bm1.1  2022  reu2  2723
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