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Theorem 2mo 2538
Description: Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
Assertion
Ref Expression
2mo (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2mo
StepHypRef Expression
1 2mo2 2537 . . . 4 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
2 nfmo1 2468 . . . . . . 7 𝑥∃*𝑥𝑦𝜑
3 nfe1 2013 . . . . . . . 8 𝑥𝑥𝜑
43nfmo 2474 . . . . . . 7 𝑥∃*𝑦𝑥𝜑
52, 4nfan 1815 . . . . . 6 𝑥(∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)
6 nfe1 2013 . . . . . . . . 9 𝑦𝑦𝜑
76nfmo 2474 . . . . . . . 8 𝑦∃*𝑥𝑦𝜑
8 nfmo1 2468 . . . . . . . 8 𝑦∃*𝑦𝑥𝜑
97, 8nfan 1815 . . . . . . 7 𝑦(∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑)
10 19.8a 2038 . . . . . . . . 9 (𝜑 → ∃𝑦𝜑)
11 spsbe 1870 . . . . . . . . . 10 ([𝑤 / 𝑦]𝜑 → ∃𝑦𝜑)
1211sbimi 1872 . . . . . . . . 9 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → [𝑧 / 𝑥]∃𝑦𝜑)
13 nfv 1829 . . . . . . . . . . . 12 𝑧𝑦𝜑
1413mo3 2494 . . . . . . . . . . 11 (∃*𝑥𝑦𝜑 ↔ ∀𝑥𝑧((∃𝑦𝜑 ∧ [𝑧 / 𝑥]∃𝑦𝜑) → 𝑥 = 𝑧))
1514biimpi 204 . . . . . . . . . 10 (∃*𝑥𝑦𝜑 → ∀𝑥𝑧((∃𝑦𝜑 ∧ [𝑧 / 𝑥]∃𝑦𝜑) → 𝑥 = 𝑧))
161519.21bbi 2047 . . . . . . . . 9 (∃*𝑥𝑦𝜑 → ((∃𝑦𝜑 ∧ [𝑧 / 𝑥]∃𝑦𝜑) → 𝑥 = 𝑧))
1710, 12, 16syl2ani 685 . . . . . . . 8 (∃*𝑥𝑦𝜑 → ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → 𝑥 = 𝑧))
18 19.8a 2038 . . . . . . . . 9 (𝜑 → ∃𝑥𝜑)
19 sbcom2 2432 . . . . . . . . . 10 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
20 spsbe 1870 . . . . . . . . . . 11 ([𝑧 / 𝑥]𝜑 → ∃𝑥𝜑)
2120sbimi 1872 . . . . . . . . . 10 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 → [𝑤 / 𝑦]∃𝑥𝜑)
2219, 21sylbi 205 . . . . . . . . 9 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → [𝑤 / 𝑦]∃𝑥𝜑)
23 nfv 1829 . . . . . . . . . . . 12 𝑤𝑥𝜑
2423mo3 2494 . . . . . . . . . . 11 (∃*𝑦𝑥𝜑 ↔ ∀𝑦𝑤((∃𝑥𝜑 ∧ [𝑤 / 𝑦]∃𝑥𝜑) → 𝑦 = 𝑤))
2524biimpi 204 . . . . . . . . . 10 (∃*𝑦𝑥𝜑 → ∀𝑦𝑤((∃𝑥𝜑 ∧ [𝑤 / 𝑦]∃𝑥𝜑) → 𝑦 = 𝑤))
262519.21bbi 2047 . . . . . . . . 9 (∃*𝑦𝑥𝜑 → ((∃𝑥𝜑 ∧ [𝑤 / 𝑦]∃𝑥𝜑) → 𝑦 = 𝑤))
2718, 22, 26syl2ani 685 . . . . . . . 8 (∃*𝑦𝑥𝜑 → ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → 𝑦 = 𝑤))
2817, 27anim12ii 591 . . . . . . 7 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) → ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
299, 28alrimi 2068 . . . . . 6 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) → ∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
305, 29alrimi 2068 . . . . 5 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) → ∀𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
3130alrimivv 1842 . . . 4 ((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) → ∀𝑧𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
321, 31sylbir 223 . . 3 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑧𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
33 nfs1v 2424 . . . . . . . 8 𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
34 nfs1v 2424 . . . . . . . . . 10 𝑦[𝑤 / 𝑦]𝜑
3534nfsb 2427 . . . . . . . . 9 𝑦[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
36 pm3.21 462 . . . . . . . . . 10 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (𝜑 → (𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)))
3736imim1d 79 . . . . . . . . 9 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
3835, 37alimd 2067 . . . . . . . 8 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (∀𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
3933, 38alimd 2067 . . . . . . 7 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
4039com12 32 . . . . . 6 (∀𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
4140aleximi 1748 . . . . 5 (∀𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
4241aleximi 1748 . . . 4 (∀𝑧𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
43 2nexaln 1746 . . . . . 6 (¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
44 2sb8e 2454 . . . . . 6 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4543, 44xchnxbi 320 . . . . 5 (¬ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
46 pm2.21 118 . . . . . . . . 9 𝜑 → (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
47462alimi 1730 . . . . . . . 8 (∀𝑥𝑦 ¬ 𝜑 → ∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
48472eximi 1752 . . . . . . 7 (∃𝑧𝑤𝑥𝑦 ¬ 𝜑 → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
494819.23bi 2048 . . . . . 6 (∃𝑤𝑥𝑦 ¬ 𝜑 → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
504919.23bi 2048 . . . . 5 (∀𝑥𝑦 ¬ 𝜑 → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
5145, 50sylbi 205 . . . 4 (¬ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
5242, 51pm2.61d1 169 . . 3 (∀𝑧𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
5332, 52impbii 197 . 2 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
54 alrot4 2025 . 2 (∀𝑧𝑤𝑥𝑦((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
5553, 54bitri 262 1 (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472  wex 1694  [wsb 1866  ∃*wmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462
This theorem is referenced by:  2mos  2539
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