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Theorem mor 2080
Description: Converse of mo23 2079 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 1868 . 2 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 impexp 263 . . . . 5 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
4 bi2.04 248 . . . . 5 ((𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
53, 4bitri 184 . . . 4 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
652albii 1482 . . 3 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
7 nfs1v 1951 . . . . . 6 𝑥[𝑦 / 𝑥]𝜑
87nfri 1530 . . . . 5 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
98eximi 1611 . . . 4 (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥[𝑦 / 𝑥]𝜑)
10 alim 1468 . . . . . . 7 (∀𝑥([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → (∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
1110alimi 1466 . . . . . 6 (∀𝑦𝑥([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → ∀𝑦(∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
1211a7s 1465 . . . . 5 (∀𝑥𝑦([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → ∀𝑦(∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
13 exim 1610 . . . . 5 (∀𝑦(∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)) → (∃𝑦𝑥[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
1412, 13syl 14 . . . 4 (∀𝑥𝑦([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → (∃𝑦𝑥[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
159, 14syl5com 29 . . 3 (∃𝑦[𝑦 / 𝑥]𝜑 → (∀𝑥𝑦([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
166, 15biimtrid 152 . 2 (∃𝑦[𝑦 / 𝑥]𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
172, 16sylbi 121 1 (∃𝑥𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362  wnf 1471  wex 1503  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  modc  2081
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