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Theorem axc11next 39133
Description: This theorem shows that, given axext4 2755, we can derive a version of axc11n 2462. However, it is weaker than axc11n 2462 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 2751 . . . . . 6 (∀𝑤(𝑤𝑥𝑤𝑧) → 𝑥 = 𝑧)
21alimi 1887 . . . . 5 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑥 𝑥 = 𝑧)
3 ax-11 2190 . . . . . . 7 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤𝑥(𝑤𝑥𝑤𝑧))
4 ax9 2158 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
5 biimpr 210 . . . . . . . . . . 11 ((𝑤𝑥𝑤𝑧) → (𝑤𝑧𝑤𝑥))
65alimi 1887 . . . . . . . . . 10 (∀𝑥(𝑤𝑥𝑤𝑧) → ∀𝑥(𝑤𝑧𝑤𝑥))
7 stdpc5v 2019 . . . . . . . . . 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 1997 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑤𝑥(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥)))
113, 10syl5 34 . . . . . 6 (𝑥 = 𝑧 → (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥)))
1211sps 2209 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥)))
132, 12mpcom 38 . . . 4 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥))
1413axc4i 2295 . . 3 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑥𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥))
15 nfa1 2184 . . . . . . . 8 𝑥𝑥 𝑤𝑥
161519.23 2236 . . . . . . 7 (∀𝑥(𝑤𝑥 → ∀𝑥 𝑤𝑥) ↔ (∃𝑥 𝑤𝑥 → ∀𝑥 𝑤𝑥))
17 19.8a 2206 . . . . . . . . 9 (𝑤𝑧 → ∃𝑧 𝑤𝑧)
18 elequ2 2159 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑤𝑧𝑤𝑥))
1918cbvexvw 2126 . . . . . . . . 9 (∃𝑧 𝑤𝑧 ↔ ∃𝑥 𝑤𝑥)
2017, 19sylib 208 . . . . . . . 8 (𝑤𝑧 → ∃𝑥 𝑤𝑥)
214cbvalivw 2092 . . . . . . . 8 (∀𝑥 𝑤𝑥 → ∀𝑧 𝑤𝑧)
2220, 21imim12i 62 . . . . . . 7 ((∃𝑥 𝑤𝑥 → ∀𝑥 𝑤𝑥) → (𝑤𝑧 → ∀𝑧 𝑤𝑧))
2316, 22sylbi 207 . . . . . 6 (∀𝑥(𝑤𝑥 → ∀𝑥 𝑤𝑥) → (𝑤𝑧 → ∀𝑧 𝑤𝑧))
2423alimi 1887 . . . . 5 (∀𝑤𝑥(𝑤𝑥 → ∀𝑥 𝑤𝑥) → ∀𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧))
2524alcoms 2191 . . . 4 (∀𝑥𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥) → ∀𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧))
2625alrimiv 2007 . . 3 (∀𝑥𝑤(𝑤𝑥 → ∀𝑥 𝑤𝑥) → ∀𝑧𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧))
27 nfa1 2184 . . . . . . . 8 𝑧𝑧 𝑤𝑧
282719.23 2236 . . . . . . 7 (∀𝑧(𝑤𝑧 → ∀𝑧 𝑤𝑧) ↔ (∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧))
29 ax9 2158 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑤𝑧𝑤𝑥))
3029spimv 2419 . . . . . . . . 9 (∀𝑧 𝑤𝑧𝑤𝑥)
3117, 30imim12i 62 . . . . . . . 8 ((∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑧𝑤𝑥))
32 19.8a 2206 . . . . . . . . . 10 (𝑤𝑥 → ∃𝑥 𝑤𝑥)
33 elequ2 2159 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑤𝑥𝑤𝑧))
3433cbvexvw 2126 . . . . . . . . . 10 (∃𝑥 𝑤𝑥 ↔ ∃𝑧 𝑤𝑧)
3532, 34sylib 208 . . . . . . . . 9 (𝑤𝑥 → ∃𝑧 𝑤𝑧)
36 sp 2207 . . . . . . . . 9 (∀𝑧 𝑤𝑧𝑤𝑧)
3735, 36imim12i 62 . . . . . . . 8 ((∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑥𝑤𝑧))
3831, 37impbid 202 . . . . . . 7 ((∃𝑧 𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑧𝑤𝑥))
3928, 38sylbi 207 . . . . . 6 (∀𝑧(𝑤𝑧 → ∀𝑧 𝑤𝑧) → (𝑤𝑧𝑤𝑥))
4039alimi 1887 . . . . 5 (∀𝑤𝑧(𝑤𝑧 → ∀𝑧 𝑤𝑧) → ∀𝑤(𝑤𝑧𝑤𝑥))
4140alcoms 2191 . . . 4 (∀𝑧𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧) → ∀𝑤(𝑤𝑧𝑤𝑥))
4241axc4i 2295 . . 3 (∀𝑧𝑤(𝑤𝑧 → ∀𝑧 𝑤𝑧) → ∀𝑧𝑤(𝑤𝑧𝑤𝑥))
4314, 26, 423syl 18 . 2 (∀𝑥𝑤(𝑤𝑥𝑤𝑧) → ∀𝑧𝑤(𝑤𝑧𝑤𝑥))
44 axext4 2755 . . 3 (𝑥 = 𝑧 ↔ ∀𝑤(𝑤𝑥𝑤𝑧))
4544albii 1895 . 2 (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑥𝑤(𝑤𝑥𝑤𝑧))
46 axext4 2755 . . 3 (𝑧 = 𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
4746albii 1895 . 2 (∀𝑧 𝑧 = 𝑥 ↔ ∀𝑧𝑤(𝑤𝑧𝑤𝑥))
4843, 45, 473imtr4i 281 1 (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629   = wceq 1631  wex 1852  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
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