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Theorem bj-mo3OLD 32512
Description: Obsolete proof temporarily kept here to check it gives no additional insight. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
bj-mo3OLD.nf 𝑦𝜑
Assertion
Ref Expression
bj-mo3OLD (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-mo3OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mo2v 2476 . . 3 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 bj-mo3OLD.nf . . . . . . . . 9 𝑦𝜑
3 nfv 1840 . . . . . . . . 9 𝑦 𝑥 = 𝑧
42, 3nfim 1822 . . . . . . . 8 𝑦(𝜑𝑥 = 𝑧)
5 nfs1v 2436 . . . . . . . . 9 𝑥[𝑦 / 𝑥]𝜑
6 nfv 1840 . . . . . . . . 9 𝑥 𝑦 = 𝑧
75, 6nfim 1822 . . . . . . . 8 𝑥([𝑦 / 𝑥]𝜑𝑦 = 𝑧)
8 sbequ2 1879 . . . . . . . . 9 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
9 ax7 1940 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
108, 9imim12d 81 . . . . . . . 8 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
114, 7, 10cbv3 2264 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1211ancli 573 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (∀𝑥(𝜑𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
134, 7aaan 2167 . . . . . 6 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) ↔ (∀𝑥(𝜑𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
1412, 13sylibr 224 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
15 prth 594 . . . . . . 7 (((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧𝑦 = 𝑧)))
16 equtr2 1951 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
1715, 16syl6 35 . . . . . 6 (((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
18172alimi 1737 . . . . 5 (∀𝑥𝑦((𝜑𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1914, 18syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
2019exlimiv 1855 . . 3 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
211, 20sylbi 207 . 2 (∃*𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
22 nfa1 2025 . . . . . 6 𝑦𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)
23 pm3.3 460 . . . . . . . . . 10 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → ([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
2423com3r 87 . . . . . . . . 9 ([𝑦 / 𝑥]𝜑 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑𝑥 = 𝑦)))
255, 24alimd 2079 . . . . . . . 8 ([𝑦 / 𝑥]𝜑 → (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑥(𝜑𝑥 = 𝑦)))
2625com12 32 . . . . . . 7 (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
2726sps 2053 . . . . . 6 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑𝑥 = 𝑦)))
2822, 27eximd 2083 . . . . 5 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
292sb8e 2424 . . . . 5 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
302mo2 2478 . . . . 5 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3128, 29, 303imtr4g 285 . . . 4 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃*𝑥𝜑))
32 moabs 2500 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
3331, 32sylibr 224 . . 3 (∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
3433alcoms 2032 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
3521, 34impbii 199 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  wnf 1705  [wsb 1877  ∃*wmo 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474
This theorem is referenced by: (None)
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