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Theorem eu3h 1942
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
Hypothesis
Ref Expression
eu3h.1 (φyφ)
Assertion
Ref Expression
eu3h (∃!xφ ↔ (xφ yx(φx = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem eu3h
StepHypRef Expression
1 euex 1927 . . 3 (∃!xφxφ)
2 eu3h.1 . . . 4 (φyφ)
32eumo0 1928 . . 3 (∃!xφyx(φx = y))
41, 3jca 290 . 2 (∃!xφ → (xφ yx(φx = y)))
52nfi 1348 . . . . 5 yφ
65mo23 1938 . . . 4 (yx(φx = y) → xy((φ [y / x]φ) → x = y))
76anim2i 324 . . 3 ((xφ yx(φx = y)) → (xφ xy((φ [y / x]φ) → x = y)))
85eu2 1941 . . 3 (∃!xφ ↔ (xφ xy((φ [y / x]φ) → x = y)))
97, 8sylibr 137 . 2 ((xφ yx(φx = y)) → ∃!xφ)
104, 9impbii 117 1 (∃!xφ ↔ (xφ yx(φx = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  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:  eu3  1943  mo2r  1949  2eu4  1990
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