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Theorem ax6e2ndALT 38188
 Description: If at least two sets exist (dtru 4783) , then the same is true expressed in an alternate form similar to the form of ax6e 2238. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndVD 38166. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2ndALT (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣

Proof of Theorem ax6e2ndALT
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . . . . 7 𝑢 ∈ V
2 ax6e 2238 . . . . . . 7 𝑦 𝑦 = 𝑣
31, 2pm3.2i 470 . . . . . 6 (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4 19.42v 1905 . . . . . . 7 (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
54biimpri 217 . . . . . 6 ((𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) → ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣))
63, 5ax-mp 5 . . . . 5 𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
7 isset 3180 . . . . . . 7 (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
87anbi1i 727 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢𝑦 = 𝑣))
98exbii 1764 . . . . 5 (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣))
106, 9mpbi 219 . . . 4 𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣)
11 id 22 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12 hbnae 2305 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13 hbn1 2007 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14 ax-5 1827 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15 ax-5 1827 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦𝑧 = 𝑦)
17 equequ1 1939 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
1817a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝑦𝑧 = 𝑦) → (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣)))
1916, 18ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
2014, 15, 19dvelimh 2324 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2111, 20syl 17 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2221idiALT 37704 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2322alimi 1730 . . . . . . . . . . . 12 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2413, 23syl 17 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2511, 24syl 17 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
26 19.41rg 37787 . . . . . . . . . 10 (∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣) → ((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
2725, 26syl 17 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
2827idiALT 37704 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
2928alimi 1730 . . . . . . 7 (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
3012, 29syl 17 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
3111, 30syl 17 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
32 exim 1751 . . . . 5 (∀𝑦((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)) → (∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
3331, 32syl 17 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
34 pm3.35 609 . . . 4 ((∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) ∧ (∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
3510, 33, 34sylancr 694 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
36 excomim 2030 . . 3 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
3735, 36syl 17 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
3837idiALT 37704 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173 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-v 3175 This theorem is referenced by: (None)
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