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Theorem kmlem10 10103
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 10102 . 2 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
3 vex 3451 . . . . 5 𝑥 ∈ V
43abrexex 7899 . . . 4 {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ∈ V
51, 4eqeltri 2830 . . 3 𝐴 ∈ V
6 raleq 3308 . . . . 5 ( = 𝐴 → (∀𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
76raleqbi1dv 3306 . . . 4 ( = 𝐴 → (∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
8 raleq 3308 . . . . 5 ( = 𝐴 → (∀𝑧 𝜑 ↔ ∀𝑧𝐴 𝜑))
98exbidv 1925 . . . 4 ( = 𝐴 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑧𝐴 𝜑))
107, 9imbi12d 345 . . 3 ( = 𝐴 → ((∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) ↔ (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧𝐴 𝜑)))
115, 10spcv 3566 . 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 1540   = wceq 1542  wex 1782  {cab 2710  wne 2940  wral 3061  wrex 3070  Vcvv 3447  cdif 3911  cin 3913  c0 4286  {csn 4590   cuni 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-rep 5246
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-sn 4591  df-uni 4870
This theorem is referenced by:  kmlem13  10106
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