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Theorem ax6e2nd 43319
Description: If at least two sets exist (dtru 5437), then the same is true expressed in an alternate form similar to the form of ax6e 2383. ax6e2nd 43319 is derived from ax6e2ndVD 43669. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2nd (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣

Proof of Theorem ax6e2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3479 . . . . . . 7 𝑢 ∈ V
2 ax6e 2383 . . . . . . 7 𝑦 𝑦 = 𝑣
31, 2pm3.2i 472 . . . . . 6 (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4 19.42v 1958 . . . . . . 7 (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
54biimpri 227 . . . . . 6 ((𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) → ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣))
63, 5ax-mp 5 . . . . 5 𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
7 isset 3488 . . . . . . 7 (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
87anbi1i 625 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢𝑦 = 𝑣))
98exbii 1851 . . . . 5 (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣))
106, 9mpbi 229 . . . 4 𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣)
11 id 22 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12 hbnae 2432 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13 hbn1 2139 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14 ax-5 1914 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15 ax-5 1914 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16 id 22 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦𝑧 = 𝑦)
17 equequ1 2029 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
1816, 17syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
1918idiALT 43238 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
2014, 15, 19dvelimh 2450 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2111, 20syl 17 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2221idiALT 43238 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2322alimi 1814 . . . . . . . . . . . 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 43311 . . . . . . . . . 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 43238 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
2928alimi 1814 . . . . . . 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 1837 . . . . 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 pm2.27 42 . . . 4 (∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ((∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)) → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
3510, 33, 34mpsyl 68 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
36 excomim 2164 . . 3 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
3735, 36syl 17 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
3837idiALT 43238 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  Vcvv 3475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477
This theorem is referenced by:  ax6e2ndeq  43320  ax6e2ndeqVD  43670  ax6e2ndeqALT  43692
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