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Theorem bnj984 30765
Description: Technical lemma for bnj69 30821. 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 3429 . . 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 2690 . . 3 (𝐺𝐵𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)})
41, 3syl6rbbr 279 . 2 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)))
5 df-rex 2913 . . . 4 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
6 bnj984.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
7 bnj252 30511 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
86, 7bitri 264 . . . 4 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
95, 8bnj133 30536 . . 3 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝜒)
109sbcbii 3477 . 2 ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ [𝐺 / 𝑓]𝑛𝜒)
114, 10syl6bb 276 1 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908  [wsbc 3421   Fn wfn 5847  w-bnj17 30494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-rex 2913  df-v 3191  df-sbc 3422  df-bnj17 30495
This theorem is referenced by:  bnj985  30766
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