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Theorem mo23 1986
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 1464 . . . . 5 𝑦 𝑥 = 𝑧
31, 2nfim 1507 . . . 4 𝑦(𝜑𝑥 = 𝑧)
43nfal 1511 . . 3 𝑦𝑥(𝜑𝑥 = 𝑧)
5 nfv 1464 . . 3 𝑧𝑥(𝜑𝑥 = 𝑦)
6 equequ2 1643 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
76imbi2d 228 . . . 4 (𝑧 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜑𝑥 = 𝑦)))
87albidv 1749 . . 3 (𝑧 = 𝑦 → (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
94, 5, 8cbvex 1683 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
10 nfs1v 1860 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
11 nfv 1464 . . . . . . . 8 𝑥 𝑦 = 𝑧
1210, 11nfim 1507 . . . . . . 7 𝑥([𝑦 / 𝑥]𝜑𝑦 = 𝑧)
13 sbequ2 1696 . . . . . . . 8 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
14 ax-8 1438 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
1513, 14imim12d 73 . . . . . . 7 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
163, 12, 15cbv3 1674 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1716ancli 316 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
183nfri 1455 . . . . . 6 ((𝜑𝑥 = 𝑧) → ∀𝑦(𝜑𝑥 = 𝑧))
1912nfri 1455 . . . . . 6 (([𝑦 / 𝑥]𝜑𝑦 = 𝑧) → ∀𝑥([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
2018, 19aaanh 1521 . . . . 5 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
2117, 20sylibr 132 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
22 prth 336 . . . . . 6 (((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧𝑦 = 𝑧)))
23 equtr2 1641 . . . . . 6 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
2422, 23syl6 33 . . . . 5 (((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
25242alimi 1388 . . . 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 1532 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
289, 27sylbir 133 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1285  wnf 1392  wex 1424  [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690
This theorem is referenced by:  modc  1988  eu2  1989  eu3h  1990
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