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Theorem modc 1940
Description: Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
Hypothesis
Ref Expression
modc.1 yφ
Assertion
Ref Expression
modc (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem modc
StepHypRef Expression
1 modc.1 . . 3 yφ
21mo23 1938 . 2 (yx(φx = y) → xy((φ [y / x]φ) → x = y))
3 exmiddc 743 . . 3 (DECID xφ → (xφ ¬ xφ))
41mor 1939 . . . 4 (xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
51mo2n 1925 . . . . 5 xφyx(φx = y))
65a1d 22 . . . 4 xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
74, 6jaoi 635 . . 3 ((xφ ¬ xφ) → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
83, 7syl 14 . 2 (DECID xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
92, 8impbid2 131 1 (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741  wal 1240  wnf 1346  wex 1378  [wsb 1642
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643
This theorem is referenced by:  mo2dc  1952
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