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Theorem mor 1939
 Description: Converse of mo23 1938 with an additional ∃xφ condition. (Contributed by Jim Kingdon, 25-Jun-2018.)
Hypothesis
Ref Expression
mor.1 yφ
Assertion
Ref Expression
mor (xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem mor
StepHypRef Expression
1 mor.1 . . 3 yφ
21sb8e 1734 . 2 (xφy[y / x]φ)
3 impexp 250 . . . . 5 (((φ [y / x]φ) → x = y) ↔ (φ → ([y / x]φx = y)))
4 bi2.04 237 . . . . 5 ((φ → ([y / x]φx = y)) ↔ ([y / x]φ → (φx = y)))
53, 4bitri 173 . . . 4 (((φ [y / x]φ) → x = y) ↔ ([y / x]φ → (φx = y)))
652albii 1357 . . 3 (xy((φ [y / x]φ) → x = y) ↔ xy([y / x]φ → (φx = y)))
7 nfs1v 1812 . . . . . 6 x[y / x]φ
87nfri 1409 . . . . 5 ([y / x]φx[y / x]φ)
98eximi 1488 . . . 4 (y[y / x]φyx[y / x]φ)
10 alim 1343 . . . . . . 7 (x([y / x]φ → (φx = y)) → (x[y / x]φx(φx = y)))
1110alimi 1341 . . . . . 6 (yx([y / x]φ → (φx = y)) → y(x[y / x]φx(φx = y)))
1211a7s 1340 . . . . 5 (xy([y / x]φ → (φx = y)) → y(x[y / x]φx(φx = y)))
13 exim 1487 . . . . 5 (y(x[y / x]φx(φx = y)) → (yx[y / x]φyx(φx = y)))
1412, 13syl 14 . . . 4 (xy([y / x]φ → (φx = y)) → (yx[y / x]φyx(φx = y)))
159, 14syl5com 26 . . 3 (y[y / x]φ → (xy([y / x]φ → (φx = y)) → yx(φx = y)))
166, 15syl5bi 141 . 2 (y[y / x]φ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
172, 16sylbi 114 1 (xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  Ⅎwnf 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-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 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  modc  1940
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