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Theorem kmlem10 8864
 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 8863 . 2 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
3 vex 3176 . . . . 5 𝑥 ∈ V
43abrexex 7033 . . . 4 {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ∈ V
51, 4eqeltri 2684 . . 3 𝐴 ∈ V
6 raleq 3115 . . . . 5 ( = 𝐴 → (∀𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
76raleqbi1dv 3123 . . . 4 ( = 𝐴 → (∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
8 raleq 3115 . . . . 5 ( = 𝐴 → (∀𝑧 𝜑 ↔ ∀𝑧𝐴 𝜑))
98exbidv 1837 . . . 4 ( = 𝐴 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑧𝐴 𝜑))
107, 9imbi12d 333 . . 3 ( = 𝐴 → ((∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) ↔ (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧𝐴 𝜑)))
115, 10spcv 3272 . 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 1473   = wceq 1475  ∃wex 1695  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539  ∅c0 3874  {csn 4125  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  kmlem13  8867
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