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Theorem 2eu6 2729
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 2727 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
2 nfia1 2198 . . . . . 6 𝑥(∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
3 nfa1 2196 . . . . . . . . . . . . 13 𝑦𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))
4 nfv 2005 . . . . . . . . . . . . 13 𝑦 𝑥 = 𝑧
5 simpl 470 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑥 = 𝑧)
65imim2i 16 . . . . . . . . . . . . . 14 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑𝑥 = 𝑧))
76sps 2221 . . . . . . . . . . . . 13 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑𝑥 = 𝑧))
83, 4, 7exlimd 2255 . . . . . . . . . . . 12 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑦𝜑𝑥 = 𝑧))
9 ax12v 2216 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
108, 9syli 39 . . . . . . . . . . 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 2223 . . . . . . . . 9 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑)))
13 nfs1v 2288 . . . . . . . . . . . . 13 𝑦[𝑤 / 𝑦]𝜑
14 simpr 473 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝑦 = 𝑤)
1514imim2i 16 . . . . . . . . . . . . . . 15 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑𝑦 = 𝑤))
16 sbequ1 2279 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 → (𝜑 → [𝑤 / 𝑦]𝜑))
1715, 16syli 39 . . . . . . . . . . . . . 14 ((𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
1817sps 2221 . . . . . . . . . . . . 13 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → [𝑤 / 𝑦]𝜑))
193, 13, 18exlimd 2255 . . . . . . . . . . . 12 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑦𝜑 → [𝑤 / 𝑦]𝜑))
2019imim2d 57 . . . . . . . . . . 11 (∀𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ((𝑥 = 𝑧 → ∃𝑦𝜑) → (𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
2120al2imi 1900 . . . . . . . . . 10 (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∀𝑥(𝑥 = 𝑧 → ∃𝑦𝜑) → ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑)))
22 sb6 2286 . . . . . . . . . . 11 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
23 2sb6 2291 . . . . . . . . . . 11 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
2422, 23bitr3i 268 . . . . . . . . . 10 (∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
2521, 24syl6ib 242 . . . . . . . . 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 542 . . . . . . 7 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))))
28 2albiim 1979 . . . . . . 7 (∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑)))
2927, 28syl6ibr 243 . . . . . 6 (∃𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤))))
302, 29exlimi 2254 . . . . 5 (∃𝑥𝑦𝜑 → (∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤))))
31302eximdv 2010 . . . 4 (∃𝑥𝑦𝜑 → (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤))))
3231imp 395 . . 3 ((∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) → ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
33 biimpr 211 . . . . . . 7 ((𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
34332alimi 1897 . . . . . 6 (∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
35342eximi 1920 . . . . 5 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
36 2exsb 2632 . . . . 5 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
3735, 36sylibr 225 . . . 4 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑥𝑦𝜑)
38 biimp 206 . . . . . 6 ((𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
39382alimi 1897 . . . . 5 (∀𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∀𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
40392eximi 1920 . . . 4 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))
4137, 40jca 503 . . 3 (∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)) → (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
4232, 41impbii 200 . 2 ((∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
431, 42bitri 266 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635  wex 1859  [wsb 2061  ∃!weu 2637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638
This theorem is referenced by: (None)
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