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Theorem ax6e2ndeq 40758
Description: "At least two sets exist" expressed in the form of dtru 5268 is logically equivalent to the same expressed in a form similar to ax6e 2396 if dtru 5268 is false implies 𝑢 = 𝑣. ax6e2ndeq 40758 is derived from ax6e2ndeqVD 41108. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2ndeq ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Proof of Theorem ax6e2ndeq
StepHypRef Expression
1 ax6e2nd 40757 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 ax6e2eq 40756 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
31a1d 25 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
42, 3pm2.61i 183 . . 3 (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
51, 4jaoi 853 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
6 olc 864 . . . 4 (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
76a1d 25 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
8 excom 2161 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
9 neeq1 3083 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥𝑣𝑢𝑣))
109biimprcd 251 . . . . . . . . . . . 12 (𝑢𝑣 → (𝑥 = 𝑢𝑥𝑣))
1110adantrd 492 . . . . . . . . . . 11 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥𝑣))
12 simpr 485 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
1312a1i 11 . . . . . . . . . . 11 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣))
14 neeq2 3084 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
1514biimprcd 251 . . . . . . . . . . 11 (𝑥𝑣 → (𝑦 = 𝑣𝑥𝑦))
1611, 13, 15syl6c 70 . . . . . . . . . 10 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥𝑦))
17 sp 2174 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
1817necon3ai 3046 . . . . . . . . . 10 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
1916, 18syl6 35 . . . . . . . . 9 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
2019eximdv 1911 . . . . . . . 8 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
21 nfnae 2453 . . . . . . . . 9 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
222119.9 2198 . . . . . . . 8 (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2320, 22syl6ib 252 . . . . . . 7 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
2423eximdv 1911 . . . . . 6 (𝑢𝑣 → (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
258, 24syl5bi 243 . . . . 5 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
26 nfnae 2453 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
272619.9 2198 . . . . 5 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2825, 27syl6ib 252 . . . 4 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
29 orc 863 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
3028, 29syl6 35 . . 3 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
317, 30pm2.61ine 3105 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
325, 31impbii 210 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  wal 1528   = wceq 1530  wex 1773  wne 3021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-ne 3022  df-v 3502
This theorem is referenced by:  2sb5nd  40759  2uasbanh  40760  2sb5ndVD  41109  2uasbanhVD  41110  2sb5ndALT  41131
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