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Theorem kmlem10 10201
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 10200 . 2 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
3 vex 3483 . . . . 5 𝑥 ∈ V
43abrexex 7988 . . . 4 {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ∈ V
51, 4eqeltri 2836 . . 3 𝐴 ∈ V
6 raleq 3322 . . . . 5 ( = 𝐴 → (∀𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
76raleqbi1dv 3337 . . . 4 ( = 𝐴 → (∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
8 raleq 3322 . . . . 5 ( = 𝐴 → (∀𝑧 𝜑 ↔ ∀𝑧𝐴 𝜑))
98exbidv 1920 . . . 4 ( = 𝐴 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑧𝐴 𝜑))
107, 9imbi12d 344 . . 3 ( = 𝐴 → ((∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) ↔ (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧𝐴 𝜑)))
115, 10spcv 3604 . 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 1537   = wceq 1539  wex 1778  {cab 2713  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  cin 3949  c0 4332  {csn 4625   cuni 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-rep 5278
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333  df-sn 4626  df-uni 4907
This theorem is referenced by:  kmlem13  10204
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