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Theorem mo23 1923
 Description: An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
Hypothesis
Ref Expression
mo23.1 yφ
Assertion
Ref Expression
mo23 (yx(φx = y) → xy((φ [y / x]φ) → x = y))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem mo23
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 mo23.1 . . . . 5 yφ
2 nfv 1402 . . . . 5 y x = z
31, 2nfim 1446 . . . 4 y(φx = z)
43nfal 1450 . . 3 yx(φx = z)
5 nfv 1402 . . 3 zx(φx = y)
6 equequ2 1581 . . . . 5 (z = y → (x = zx = y))
76imbi2d 219 . . . 4 (z = y → ((φx = z) ↔ (φx = y)))
87albidv 1687 . . 3 (z = y → (x(φx = z) ↔ x(φx = y)))
94, 5, 8cbvex 1621 . 2 (zx(φx = z) ↔ yx(φx = y))
10 nfs1v 1797 . . . . . . . 8 x[y / x]φ
11 nfv 1402 . . . . . . . 8 x y = z
1210, 11nfim 1446 . . . . . . 7 x([y / x]φy = z)
13 sbequ2 1634 . . . . . . . 8 (x = y → ([y / x]φφ))
14 ax-8 1376 . . . . . . . 8 (x = y → (x = zy = z))
1513, 14imim12d 68 . . . . . . 7 (x = y → ((φx = z) → ([y / x]φy = z)))
163, 12, 15cbv3 1612 . . . . . 6 (x(φx = z) → y([y / x]φy = z))
1716ancli 306 . . . . 5 (x(φx = z) → (x(φx = z) y([y / x]φy = z)))
183nfri 1393 . . . . . 6 ((φx = z) → y(φx = z))
1912nfri 1393 . . . . . 6 (([y / x]φy = z) → x([y / x]φy = z))
2018, 19aaanh 1460 . . . . 5 (xy((φx = z) ([y / x]φy = z)) ↔ (x(φx = z) y([y / x]φy = z)))
2117, 20sylibr 137 . . . 4 (x(φx = z) → xy((φx = z) ([y / x]φy = z)))
22 prth 326 . . . . . 6 (((φx = z) ([y / x]φy = z)) → ((φ [y / x]φ) → (x = z y = z)))
23 equtr2 1579 . . . . . 6 ((x = z y = z) → x = y)
2422, 23syl6 29 . . . . 5 (((φx = z) ([y / x]φy = z)) → ((φ [y / x]φ) → x = y))
25242alimi 1325 . . . 4 (xy((φx = z) ([y / x]φy = z)) → xy((φ [y / x]φ) → x = y))
2621, 25syl 14 . . 3 (x(φx = z) → xy((φ [y / x]φ) → x = y))
2726exlimiv 1471 . 2 (zx(φx = z) → xy((φ [y / x]φ) → x = y))
289, 27sylbir 125 1 (yx(φx = y) → xy((φ [y / x]φ) → x = y))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362  [wsb 1627 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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628 This theorem is referenced by:  modc  1925  eu2  1926  eu3h  1927
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