Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax6e2nd Structured version   Visualization version   GIF version

Theorem ax6e2nd 39293
 Description: If at least two sets exist (dtru 4985) , then the same is true expressed in an alternate form similar to the form of ax6e 2411. ax6e2nd 39293 is derived from ax6e2ndVD 39660. (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 3352 . . . . . . 7 𝑢 ∈ V
2 ax6e 2411 . . . . . . 7 𝑦 𝑦 = 𝑣
31, 2pm3.2i 447 . . . . . 6 (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4 19.42v 2032 . . . . . . 7 (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
54biimpri 218 . . . . . 6 ((𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) → ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣))
63, 5ax-mp 5 . . . . 5 𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
7 isset 3356 . . . . . . 7 (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
87anbi1i 602 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢𝑦 = 𝑣))
98exbii 1923 . . . . 5 (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣))
106, 9mpbi 220 . . . 4 𝑦(∃𝑥 𝑥 = 𝑢𝑦 = 𝑣)
11 id 22 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12 hbnae 2468 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13 hbn1 2174 . . . . . . . . . . . 12 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14 ax-5 1990 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15 ax-5 1990 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16 id 22 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦𝑧 = 𝑦)
17 equequ1 2109 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
1816, 17syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
1918idiALT 39202 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (𝑧 = 𝑣𝑦 = 𝑣))
2014, 15, 19dvelimh 2485 . . . . . . . . . . . . . . 15 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2111, 20syl 17 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2221idiALT 39202 . . . . . . . . . . . . 13 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
2322alimi 1886 . . . . . . . . . . . 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 39285 . . . . . . . . . 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 39202 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢𝑦 = 𝑣)))
2928alimi 1886 . . . . . . 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 1908 . . . . 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 2198 . . 3 (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
3735, 36syl 17 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
3837idiALT 39202 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1628  ∃wex 1851   ∈ wcel 2144  Vcvv 3349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-v 3351 This theorem is referenced by:  ax6e2ndeq  39294  ax6e2ndeqVD  39661  ax6e2ndeqALT  39683
 Copyright terms: Public domain W3C validator