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Theorem mor 1958
Description: Converse of mo23 1957 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 1753 . 2 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 impexp 254 . . . . 5 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ (𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
4 bi2.04 241 . . . . 5 ((𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
53, 4bitri 177 . . . 4 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
652albii 1376 . . 3 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)))
7 nfs1v 1831 . . . . . 6 𝑥[𝑦 / 𝑥]𝜑
87nfri 1428 . . . . 5 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
98eximi 1507 . . . 4 (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥[𝑦 / 𝑥]𝜑)
10 alim 1362 . . . . . . 7 (∀𝑥([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → (∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
1110alimi 1360 . . . . . 6 (∀𝑦𝑥([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → ∀𝑦(∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
1211a7s 1359 . . . . 5 (∀𝑥𝑦([𝑦 / 𝑥]𝜑 → (𝜑𝑥 = 𝑦)) → ∀𝑦(∀𝑥[𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
13 exim 1506 . . . . 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, 15syl5bi 145 . 2 (∃𝑦[𝑦 / 𝑥]𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
172, 16sylbi 118 1 (∃𝑥𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wnf 1365  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  modc  1959
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