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Theorem bnj984 32123
Description: Technical lemma for bnj69 32179. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj984.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj984.11 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj984 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))

Proof of Theorem bnj984
StepHypRef Expression
1 sbc8g 3777 . . 3 (𝐺𝐴 → ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ 𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}))
2 bnj984.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
32eleq2i 2901 . . 3 (𝐺𝐵𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)})
41, 3syl6rbbr 291 . 2 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)))
5 df-rex 3141 . . . 4 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
6 bnj984.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
7 bnj252 31872 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
86, 7bitri 276 . . . 4 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
95, 8bnj133 31896 . . 3 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝜒)
109sbcbii 3826 . 2 ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ [𝐺 / 𝑓]𝑛𝜒)
114, 10syl6bb 288 1 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wrex 3136  [wsbc 3769   Fn wfn 6343  w-bnj17 31855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rex 3141  df-sbc 3770  df-bnj17 31856
This theorem is referenced by:  bnj985  32124
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