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Theorem kmlem10 9171
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 9170 . 2 𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)
3 vex 3341 . . . . 5 𝑥 ∈ V
43abrexex 7304 . . . 4 {𝑢 ∣ ∃𝑡𝑥 𝑢 = (𝑡 (𝑥 ∖ {𝑡}))} ∈ V
51, 4eqeltri 2833 . . 3 𝐴 ∈ V
6 raleq 3275 . . . . 5 ( = 𝐴 → (∀𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
76raleqbi1dv 3283 . . . 4 ( = 𝐴 → (∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) ↔ ∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅)))
8 raleq 3275 . . . . 5 ( = 𝐴 → (∀𝑧 𝜑 ↔ ∀𝑧𝐴 𝜑))
98exbidv 1997 . . . 4 ( = 𝐴 → (∃𝑦𝑧 𝜑 ↔ ∃𝑦𝑧𝐴 𝜑))
107, 9imbi12d 333 . . 3 ( = 𝐴 → ((∀𝑧𝑤 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧 𝜑) ↔ (∀𝑧𝐴𝑤𝐴 (𝑧𝑤 → (𝑧𝑤) = ∅) → ∃𝑦𝑧𝐴 𝜑)))
115, 10spcv 3437 . 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 1628   = wceq 1630  wex 1851  {cab 2744  wne 2930  wral 3048  wrex 3049  Vcvv 3338  cdif 3710  cin 3712  c0 4056  {csn 4319   cuni 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055
This theorem is referenced by:  kmlem13  9174
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