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Theorem kmlem14 9273
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.)
Hypotheses
Ref Expression
kmlem14.1 (𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))
kmlem14.2 (𝜓 ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))
kmlem14.3 (𝜒 ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
Assertion
Ref Expression
kmlem14 (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑦𝑧𝑣𝑢(𝑦𝑥𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢   𝜑,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem kmlem14
StepHypRef Expression
1 neeq1 3033 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑤𝑦𝑤))
2 ineq1 4005 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑤) = (𝑦𝑤))
32eleq2d 2864 . . . . . 6 (𝑧 = 𝑦 → (𝑣 ∈ (𝑧𝑤) ↔ 𝑣 ∈ (𝑦𝑤)))
41, 3anbi12d 625 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ (𝑦𝑤𝑣 ∈ (𝑦𝑤))))
54rexbidv 3233 . . . 4 (𝑧 = 𝑦 → (∃𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤))))
65raleqbi1dv 3329 . . 3 (𝑧 = 𝑦 → (∀𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤))))
76cbvrexv 3355 . 2 (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑦𝑥𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤)))
8 df-rex 3095 . 2 (∃𝑦𝑥𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤)) ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤))))
9 eleq1w 2861 . . . . . . . . 9 (𝑣 = 𝑧 → (𝑣 ∈ (𝑦𝑤) ↔ 𝑧 ∈ (𝑦𝑤)))
109anbi2d 623 . . . . . . . 8 (𝑣 = 𝑧 → ((𝑦𝑤𝑣 ∈ (𝑦𝑤)) ↔ (𝑦𝑤𝑧 ∈ (𝑦𝑤))))
1110rexbidv 3233 . . . . . . 7 (𝑣 = 𝑧 → (∃𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤)) ↔ ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤))))
1211cbvralv 3354 . . . . . 6 (∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤)) ↔ ∀𝑧𝑦𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))
13 df-ral 3094 . . . . . 6 (∀𝑧𝑦𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤))))
1412, 13bitri 267 . . . . 5 (∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤)) ↔ ∀𝑧(𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤))))
1514anbi2i 617 . . . 4 ((𝑦𝑥 ∧ ∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤))) ↔ (𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))))
16 19.28v 2091 . . . 4 (∀𝑧(𝑦𝑥 ∧ (𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))) ↔ (𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))))
17 neeq2 3034 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝑦𝑤𝑦𝑣))
18 ineq2 4006 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → (𝑦𝑤) = (𝑦𝑣))
1918eleq2d 2864 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝑧 ∈ (𝑦𝑤) ↔ 𝑧 ∈ (𝑦𝑣)))
2017, 19anbi12d 625 . . . . . . . . . . 11 (𝑤 = 𝑣 → ((𝑦𝑤𝑧 ∈ (𝑦𝑤)) ↔ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))
2120cbvrexv 3355 . . . . . . . . . 10 (∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)) ↔ ∃𝑣𝑥 (𝑦𝑣𝑧 ∈ (𝑦𝑣)))
22 df-rex 3095 . . . . . . . . . 10 (∃𝑣𝑥 (𝑦𝑣𝑧 ∈ (𝑦𝑣)) ↔ ∃𝑣(𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))
2321, 22bitri 267 . . . . . . . . 9 (∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)) ↔ ∃𝑣(𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))
2423imbi2i 328 . . . . . . . 8 ((𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤))) ↔ (𝑧𝑦 → ∃𝑣(𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))))
25 19.37v 2092 . . . . . . . 8 (∃𝑣(𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))) ↔ (𝑧𝑦 → ∃𝑣(𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))))
2624, 25bitr4i 270 . . . . . . 7 ((𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤))) ↔ ∃𝑣(𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))))
2726anbi2i 617 . . . . . 6 ((𝑦𝑥 ∧ (𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))) ↔ (𝑦𝑥 ∧ ∃𝑣(𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))))
28 19.42v 2049 . . . . . 6 (∃𝑣(𝑦𝑥 ∧ (𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))) ↔ (𝑦𝑥 ∧ ∃𝑣(𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))))
29 19.3v 2081 . . . . . . . 8 (∀𝑢(𝑦𝑥𝜑) ↔ (𝑦𝑥𝜑))
30 kmlem14.1 . . . . . . . . . 10 (𝜑 ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))
31 elin 3994 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑦𝑣) ↔ (𝑧𝑦𝑧𝑣))
3231baibr 533 . . . . . . . . . . . . 13 (𝑧𝑦 → (𝑧𝑣𝑧 ∈ (𝑦𝑣)))
3332anbi2d 623 . . . . . . . . . . . 12 (𝑧𝑦 → (((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣) ↔ ((𝑣𝑥𝑦𝑣) ∧ 𝑧 ∈ (𝑦𝑣))))
34 anass 461 . . . . . . . . . . . 12 (((𝑣𝑥𝑦𝑣) ∧ 𝑧 ∈ (𝑦𝑣)) ↔ (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))
3533, 34syl6bb 279 . . . . . . . . . . 11 (𝑧𝑦 → (((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣) ↔ (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))))
3635pm5.74i 263 . . . . . . . . . 10 ((𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)) ↔ (𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))))
3730, 36bitri 267 . . . . . . . . 9 (𝜑 ↔ (𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣)))))
3837anbi2i 617 . . . . . . . 8 ((𝑦𝑥𝜑) ↔ (𝑦𝑥 ∧ (𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))))
3929, 38bitr2i 268 . . . . . . 7 ((𝑦𝑥 ∧ (𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))) ↔ ∀𝑢(𝑦𝑥𝜑))
4039exbii 1944 . . . . . 6 (∃𝑣(𝑦𝑥 ∧ (𝑧𝑦 → (𝑣𝑥 ∧ (𝑦𝑣𝑧 ∈ (𝑦𝑣))))) ↔ ∃𝑣𝑢(𝑦𝑥𝜑))
4127, 28, 403bitr2i 291 . . . . 5 ((𝑦𝑥 ∧ (𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))) ↔ ∃𝑣𝑢(𝑦𝑥𝜑))
4241albii 1915 . . . 4 (∀𝑧(𝑦𝑥 ∧ (𝑧𝑦 → ∃𝑤𝑥 (𝑦𝑤𝑧 ∈ (𝑦𝑤)))) ↔ ∀𝑧𝑣𝑢(𝑦𝑥𝜑))
4315, 16, 423bitr2i 291 . . 3 ((𝑦𝑥 ∧ ∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤))) ↔ ∀𝑧𝑣𝑢(𝑦𝑥𝜑))
4443exbii 1944 . 2 (∃𝑦(𝑦𝑥 ∧ ∀𝑣𝑦𝑤𝑥 (𝑦𝑤𝑣 ∈ (𝑦𝑤))) ↔ ∃𝑦𝑧𝑣𝑢(𝑦𝑥𝜑))
457, 8, 443bitri 289 1 (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ↔ ∃𝑦𝑧𝑣𝑢(𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651  wex 1875  wcel 2157  ∃!weu 2608  wne 2971  wral 3089  wrex 3090  cin 3768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-in 3776
This theorem is referenced by:  kmlem16  9275
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