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Theorem eu2 1922
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 1908 . . 3 (∃!xφxφ)
2 eu2.1 . . . . . 6 yφ
32nfri 1389 . . . . 5 (φyφ)
43eumo0 1909 . . . 4 (∃!xφyx(φx = y))
52mo23 1919 . . . 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 1490 . . . 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 1335 . . . . . . . 8 (y((φ [y / x]φ) → x = y) ↔ y(φ → ([y / x]φx = y)))
11219.21 1453 . . . . . . . 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 433 . . . . . 6 ((φ y((φ [y / x]φ) → x = y)) ↔ (φ (φy([y / x]φx = y))))
14 abai 482 . . . . . 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 1474 . . . 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 1903 . . 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 1224  wnf 1325  wex 1358  [wsb 1623  ∃!weu 1878
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-eu 1881
This theorem is referenced by:  eu3h  1923  mo3h  1931  bm1.1  2003  reu2  2702
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