Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axc11next Structured version   Visualization version   GIF version

Theorem axc11next 39100
Description: This theorem shows that, given axext4 2785, we can derive a version of axc11n 2473. However, it is weaker than axc11n 2473 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11next (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Distinct variable group:   𝑥,𝑧

Proof of Theorem axc11next
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-ext 2781 . . . . . 6 (∀𝑤(𝑤𝑥𝑤𝑧) → 𝑥 = 𝑧)
21alimi 1896 . . . . 5 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑥 𝑥 = 𝑧)
3 ax-11 2200 . . . . . . 7 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤𝑥(𝑤𝑥𝑤𝑧))
4 ax9 2168 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
5 biimpr 211 . . . . . . . . . . 11 ((𝑤𝑥𝑤𝑧) → (𝑤𝑧𝑤𝑥))
65alimi 1896 . . . . . . . . . 10 (∀𝑥(𝑤𝑥𝑤𝑧) → ∀𝑥(𝑤𝑧𝑤𝑥))
7 stdpc5v 2029 . . . . . . . . . 10 (∀𝑥(𝑤𝑧𝑤𝑥) → (𝑤𝑧 → ∀𝑥 𝑤𝑥))
86, 7syl 17 . . . . . . . . 9 (∀𝑥(𝑤𝑥𝑤𝑧) → (𝑤𝑧 → ∀𝑥 𝑤𝑥))
94, 8syl9 77 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑥(𝑤𝑥𝑤𝑧) → (𝑤𝑥 → ∀𝑥 𝑤𝑥)))
109alimdv 2007 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑤𝑥(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥)))
113, 10syl5 34 . . . . . 6 (𝑥 = 𝑧 → (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥)))
1211sps 2219 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥)))
132, 12mpcom 38 . . . 4 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥))
1413axc4i 2306 . . 3 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑥𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥))
15 nfa1 2194 . . . . . . . 8 𝑥𝑥 𝑤𝑥
161519.23 2246 . . . . . . 7 (∀𝑥(𝑤𝑥 → ∀𝑥 𝑤𝑥) ↔ (∃𝑥 𝑤𝑥 → ∀𝑥 𝑤𝑥))
17 19.8a 2216 . . . . . . . . 9 (𝑤𝑧 → ∃𝑧 𝑤𝑧)
18 elequ2 2169 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑤𝑧𝑤𝑥))
1918cbvexvw 2136 . . . . . . . . 9 (∃𝑧 𝑤𝑧 ↔ ∃𝑥 𝑤𝑥)
2017, 19sylib 209 . . . . . . . 8 (𝑤𝑧 → ∃𝑥 𝑤𝑥)
214cbvalivw 2102 . . . . . . . 8 (∀𝑥 𝑤𝑥 → ∀𝑧 𝑤𝑧)
2220, 21imim12i 62 . . . . . . 7 ((∃𝑥 𝑤𝑥 → ∀𝑥 𝑤𝑥) → (𝑤𝑧 → ∀𝑧 𝑤𝑧))
2316, 22sylbi 208 . . . . . 6 (∀𝑥(𝑤𝑥 → ∀𝑥 𝑤𝑥) → (𝑤𝑧 → ∀𝑧 𝑤𝑧))
2423alimi 1896 . . . . 5 (∀𝑤𝑥(𝑤𝑥 → ∀𝑥 𝑤𝑥) → ∀𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧))
2524alcoms 2201 . . . 4 (∀𝑥𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥) → ∀𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧))
2625alrimiv 2017 . . 3 (∀𝑥𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥) → ∀𝑧𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧))
27 nfa1 2194 . . . . . . . 8 𝑧𝑧 𝑤𝑧
282719.23 2246 . . . . . . 7 (∀𝑧(𝑤𝑧 → ∀𝑧 𝑤𝑧) ↔ (∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧))
29 ax9 2168 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑤𝑧𝑤𝑥))
3029spimvw 2095 . . . . . . . . 9 (∀𝑧 𝑤𝑧𝑤𝑥)
3117, 30imim12i 62 . . . . . . . 8 ((∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑧𝑤𝑥))
32 19.8a 2216 . . . . . . . . . 10 (𝑤𝑥 → ∃𝑥 𝑤𝑥)
33 elequ2 2169 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
3433cbvexvw 2136 . . . . . . . . . 10 (∃𝑥 𝑤𝑥 ↔ ∃𝑧 𝑤𝑧)
3532, 34sylib 209 . . . . . . . . 9 (𝑤𝑥 → ∃𝑧 𝑤𝑧)
36 sp 2217 . . . . . . . . 9 (∀𝑧 𝑤𝑧𝑤𝑧)
3735, 36imim12i 62 . . . . . . . 8 ((∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑥𝑤𝑧))
3831, 37impbid 203 . . . . . . 7 ((∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑧𝑤𝑥))
3928, 38sylbi 208 . . . . . 6 (∀𝑧(𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑧𝑤𝑥))
4039alimi 1896 . . . . 5 (∀𝑤𝑧(𝑤𝑧 → ∀𝑧 𝑤𝑧) → ∀𝑤(𝑤𝑧𝑤𝑥))
4140alcoms 2201 . . . 4 (∀𝑧𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧) → ∀𝑤(𝑤𝑧𝑤𝑥))
4241axc4i 2306 . . 3 (∀𝑧𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧) → ∀𝑧𝑤(𝑤𝑧𝑤𝑥))
4314, 26, 423syl 18 . 2 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑧𝑤(𝑤𝑧𝑤𝑥))
44 axext4 2785 . . 3 (𝑥 = 𝑧 ↔ ∀𝑤(𝑤𝑥𝑤𝑧))
4544albii 1904 . 2 (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑥𝑤(𝑤𝑥𝑤𝑧))
46 axext4 2785 . . 3 (𝑧 = 𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
4746albii 1904 . 2 (∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧𝑤(𝑤𝑧𝑤𝑥))
4843, 45, 473imtr4i 283 1 (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635   = wceq 1637  wex 1859  wcel 2155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator