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Theorem bnj984 32284
Description: Technical lemma for bnj69 32342. 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 3766 . . 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 2907 . . 3 (𝐺𝐵𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)})
41, 3syl6rbbr 293 . 2 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)))
5 df-rex 3139 . . . 4 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
6 bnj984.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
7 bnj252 32033 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
86, 7bitri 278 . . . 4 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
95, 8bnj133 32057 . . 3 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝜒)
109sbcbii 3814 . 2 ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ [𝐺 / 𝑓]𝑛𝜒)
114, 10syl6bb 290 1 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wrex 3134  [wsbc 3758   Fn wfn 6338  w-bnj17 32016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rex 3139  df-sbc 3759  df-bnj17 32017
This theorem is referenced by:  bnj985v  32285  bnj985  32286
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