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Theorem kmlem10 10198
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Hypothesis
Ref Expression
kmlem9.1 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
Assertion
Ref Expression
kmlem10 (∀(∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) → ∃𝑦𝑧𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑢,𝑡,   𝑦,𝐴,𝑧,𝑤,   𝜑,
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑢,𝑡)   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem10
StepHypRef Expression
1 kmlem9.1 . . 3 𝐴 = {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))}
21kmlem9 10197 . 2 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
3 vex 3482 . . . . 5 𝑥 ∈ V
43abrexex 7986 . . . 4 {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ∈ V
51, 4eqeltri 2835 . . 3 𝐴 ∈ V
6 raleq 3321 . . . . 5 ( = 𝐴 → (∀𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
76raleqbi1dv 3336 . . . 4 ( = 𝐴 → (∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
8 raleq 3321 . . . . 5 ( = 𝐴 → (∀𝑧 𝜑 ↔ ∀𝑧𝐴 𝜑))
98exbidv 1919 . . . 4 ( = 𝐴 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑧𝐴 𝜑))
107, 9imbi12d 344 . . 3 ( = 𝐴 → ((∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) ↔ (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧𝐴 𝜑)))
115, 10spcv 3605 . 2 (∀(∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) → (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧𝐴 𝜑))
122, 11mpi 20 1 (∀(∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) → ∃𝑦𝑧𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  wex 1776  {cab 2712  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cin 3962  c0 4339  {csn 4631   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-rep 5285
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913
This theorem is referenced by:  kmlem13  10201
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