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Theorem 2eu6 2546
 Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
Assertion
Ref Expression
2eu6 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu6
StepHypRef Expression
1 2eu4 2544 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
2 nfia1 2017 . . . . . 6 𝑥(∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
3 nfa1 2015 . . . . . . . . . . . . 13 𝑦𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))
4 nfv 1830 . . . . . . . . . . . . 13 𝑦 𝑥 = 𝑧
5 simpl 472 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑥 = 𝑧)
65imim2i 16 . . . . . . . . . . . . . 14 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑𝑥 = 𝑧))
76sps 2043 . . . . . . . . . . . . 13 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑𝑥 = 𝑧))
83, 4, 7exlimd 2074 . . . . . . . . . . . 12 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑦𝜑𝑥 = 𝑧))
9 ax12v 2035 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
108, 9syli 38 . . . . . . . . . . 11 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
1110com12 32 . . . . . . . . . 10 (∃𝑦𝜑 → (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
1211spsd 2045 . . . . . . . . 9 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
13 nfs1v 2425 . . . . . . . . . . . . 13 𝑦[𝑤 / 𝑦]𝜑
14 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑦 = 𝑤)
1514imim2i 16 . . . . . . . . . . . . . . 15 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑𝑦 = 𝑤))
16 sbequ1 2096 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 → (𝜑 → [𝑤 / 𝑦]𝜑))
1715, 16syli 38 . . . . . . . . . . . . . 14 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
1817sps 2043 . . . . . . . . . . . . 13 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
193, 13, 18exlimd 2074 . . . . . . . . . . . 12 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑦𝜑 → [𝑤 / 𝑦]𝜑))
2019imim2d 55 . . . . . . . . . . 11 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ((𝑥 = 𝑧 → ∃𝑦𝜑) → (𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
2120al2imi 1733 . . . . . . . . . 10 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
22 sb6 2417 . . . . . . . . . . 11 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
23 2sb6 2432 . . . . . . . . . . 11 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
2422, 23bitr3i 265 . . . . . . . . . 10 (∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
2521, 24syl6ib 240 . . . . . . . . 9 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑) → ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑)))
2612, 25sylcom 30 . . . . . . . 8 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑)))
2726ancld 574 . . . . . . 7 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))))
28 2albiim 1807 . . . . . . 7 (∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑)))
2927, 28syl6ibr 241 . . . . . 6 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤))))
302, 29exlimi 2073 . . . . 5 (∃𝑥𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤))))
31302eximdv 1835 . . . 4 (∃𝑥𝑦𝜑 → (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤))))
3231imp 444 . . 3 ((∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
33 biimpr 209 . . . . . . 7 ((𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
34332alimi 1731 . . . . . 6 (∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
35342eximi 1753 . . . . 5 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
36 2exsb 2457 . . . . 5 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
3735, 36sylibr 223 . . . 4 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑥𝑦𝜑)
38 biimp 204 . . . . . 6 ((𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
39382alimi 1731 . . . . 5 (∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
40392eximi 1753 . . . 4 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
4137, 40jca 553 . . 3 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
4232, 41impbii 198 . 2 ((∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
431, 42bitri 263 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867  ∃!weu 2458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463 This theorem is referenced by: (None)
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