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Theorem bnj984 33504
Description: Technical lemma for bnj69 33562. 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 2829 . . 3 (𝐺𝐵𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)})
3 sbc8g 3747 . . 3 (𝐺𝐴 → ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ 𝐺 ∈ {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}))
42, 3bitr4id 289 . 2 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)))
5 df-rex 3074 . . . 4 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
6 bnj984.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
7 bnj252 33255 . . . . 5 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
86, 7bitri 274 . . . 4 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
95, 8bnj133 33279 . . 3 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ ∃𝑛𝜒)
109sbcbii 3799 . 2 ([𝐺 / 𝑓]𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) ↔ [𝐺 / 𝑓]𝑛𝜒)
114, 10bitrdi 286 1 (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wrex 3073  [wsbc 3739   Fn wfn 6491  w-bnj17 33238
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3074  df-sbc 3740  df-bnj17 33239
This theorem is referenced by:  bnj985v  33505  bnj985  33506
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