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Theorem ax6e2ndeq 37796
 Description: "At least two sets exist" expressed in the form of dtru 4783 is logically equivalent to the same expressed in a form similar to ax6e 2238 if dtru 4783 is false implies 𝑢 = 𝑣. ax6e2ndeq 37796 is derived from ax6e2ndeqVD 38167. (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 37795 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 ax6e2eq 37794 . . . 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 175 . . 3 (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
51, 4jaoi 393 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
6 olc 398 . . . 4 (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
76a1d 25 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
8 excom 2029 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
9 neeq1 2844 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥𝑣𝑢𝑣))
109biimprcd 239 . . . . . . . . . . . 12 (𝑢𝑣 → (𝑥 = 𝑢𝑥𝑣))
1110adantrd 483 . . . . . . . . . . 11 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥𝑣))
12 simpr 476 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
1312a1i 11 . . . . . . . . . . 11 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣))
14 neeq2 2845 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
1514biimprcd 239 . . . . . . . . . . 11 (𝑥𝑣 → (𝑦 = 𝑣𝑥𝑦))
1611, 13, 15syl6c 68 . . . . . . . . . 10 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥𝑦))
17 sp 2041 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
1817necon3ai 2807 . . . . . . . . . 10 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
1916, 18syl6 34 . . . . . . . . 9 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
2019eximdv 1833 . . . . . . . 8 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
21 nfnae 2306 . . . . . . . . 9 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
222119.9 2060 . . . . . . . 8 (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2320, 22syl6ib 240 . . . . . . 7 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
2423eximdv 1833 . . . . . 6 (𝑢𝑣 → (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
258, 24syl5bi 231 . . . . 5 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
26 nfnae 2306 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
272619.9 2060 . . . . 5 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2825, 27syl6ib 240 . . . 4 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
29 orc 399 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
3028, 29syl6 34 . . 3 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
317, 30pm2.61ine 2865 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
325, 31impbii 198 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ≠ wne 2780 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  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-v 3175 This theorem is referenced by:  2sb5nd  37797  2uasbanh  37798  2sb5ndVD  38168  2uasbanhVD  38169  2sb5ndALT  38190
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