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Theorem ax6e2ndeq 38278
Description: "At least two sets exist" expressed in the form of dtru 4819 is logically equivalent to the same expressed in a form similar to ax6e 2249 if dtru 4819 is false implies 𝑢 = 𝑣. ax6e2ndeq 38278 is derived from ax6e2ndeqVD 38649. (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 38277 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 ax6e2eq 38276 . . . 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 176 . . 3 (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
51, 4jaoi 394 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
6 olc 399 . . . 4 (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
76a1d 25 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
8 excom 2039 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
9 neeq1 2852 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥𝑣𝑢𝑣))
109biimprcd 240 . . . . . . . . . . . 12 (𝑢𝑣 → (𝑥 = 𝑢𝑥𝑣))
1110adantrd 484 . . . . . . . . . . 11 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥𝑣))
12 simpr 477 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
1312a1i 11 . . . . . . . . . . 11 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣))
14 neeq2 2853 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
1514biimprcd 240 . . . . . . . . . . 11 (𝑥𝑣 → (𝑦 = 𝑣𝑥𝑦))
1611, 13, 15syl6c 70 . . . . . . . . . 10 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥𝑦))
17 sp 2051 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
1817necon3ai 2815 . . . . . . . . . 10 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
1916, 18syl6 35 . . . . . . . . 9 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
2019eximdv 1843 . . . . . . . 8 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
21 nfnae 2317 . . . . . . . . 9 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
222119.9 2070 . . . . . . . 8 (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2320, 22syl6ib 241 . . . . . . 7 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
2423eximdv 1843 . . . . . 6 (𝑢𝑣 → (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
258, 24syl5bi 232 . . . . 5 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
26 nfnae 2317 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
272619.9 2070 . . . . 5 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2825, 27syl6ib 241 . . . 4 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
29 orc 400 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
3028, 29syl6 35 . . 3 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
317, 30pm2.61ine 2873 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
325, 31impbii 199 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wne 2790
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-v 3188
This theorem is referenced by:  2sb5nd  38279  2uasbanh  38280  2sb5ndVD  38650  2uasbanhVD  38651  2sb5ndALT  38672
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