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Theorem bnj984 34492
Description: Technical lemma for bnj69 34550. 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 bnj984.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
21eleq2i 2819 . . 3 (𝐺𝐵𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)})
3 sbc8g 3780 . . 3 (𝐺𝐴 → ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ 𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}))
42, 3bitr4id 290 . 2 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)))
5 df-rex 3065 . . . 4 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
6 bnj984.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
7 bnj252 34243 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
86, 7bitri 275 . . . 4 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
95, 8bnj133 34267 . . 3 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝜒)
109sbcbii 3832 . 2 ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ [𝐺 / 𝑓]𝑛𝜒)
114, 10bitrdi 287 1 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wrex 3064  [wsbc 3772   Fn wfn 6531  w-bnj17 34226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-sbc 3773  df-bnj17 34227
This theorem is referenced by:  bnj985v  34493  bnj985  34494
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