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Theorem bnj985 33623
Description: Technical lemma for bnj69 33679. 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). See bnj985v 33622 for a version with more disjoint variable conditions, not requiring ax-13 2371. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj985.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj985.6 (𝜒′[𝑝 / 𝑛]𝜒)
bnj985.9 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj985.11 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj985.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj985 (𝐺𝐵 ↔ ∃𝑝𝜒″)
Distinct variable groups:   𝐺,𝑝   𝜒,𝑝   𝑓,𝑝
Allowed substitution hints:   𝜑(𝑓,𝑛,𝑝)   𝜓(𝑓,𝑛,𝑝)   𝜒(𝑓,𝑛)   𝐵(𝑓,𝑛,𝑝)   𝐶(𝑓,𝑛,𝑝)   𝐷(𝑓,𝑛,𝑝)   𝐺(𝑓,𝑛)   𝜒′(𝑓,𝑛,𝑝)   𝜒″(𝑓,𝑛,𝑝)

Proof of Theorem bnj985
StepHypRef Expression
1 bnj985.13 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
21bnj918 33435 . . 3 𝐺 ∈ V
3 bnj985.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj985.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj984 33621 . . 3 (𝐺 ∈ V → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
62, 5ax-mp 5 . 2 (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒)
7 sbcex2 3805 . . 3 ([𝐺 / 𝑓]𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8 nfv 1918 . . . . . . 7 𝑝𝜒
98sb8e 2517 . . . . . 6 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10 sbsbc 3744 . . . . . . 7 ([𝑝 / 𝑛]𝜒[𝑝 / 𝑛]𝜒)
1110exbii 1851 . . . . . 6 (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
129, 11bitri 275 . . . . 5 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13 bnj985.6 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
1412, 13bnj133 33396 . . . 4 (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
1514sbcbii 3800 . . 3 ([𝐺 / 𝑓]𝑛𝜒[𝐺 / 𝑓]𝑝𝜒′)
16 bnj985.9 . . . 4 (𝜒″[𝐺 / 𝑓]𝜒′)
1716exbii 1851 . . 3 (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
187, 15, 173bitr4i 303 . 2 ([𝐺 / 𝑓]𝑛𝜒 ↔ ∃𝑝𝜒″)
196, 18bitri 275 1 (𝐺𝐵 ↔ ∃𝑝𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1088   = wceq 1542  wex 1782  [wsb 2068  wcel 2107  {cab 2710  wrex 3070  Vcvv 3444  [wsbc 3740  cun 3909  {csn 4587  cop 4593   Fn wfn 6492  w-bnj17 33355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-sn 4588  df-pr 4590  df-uni 4867  df-bnj17 33356
This theorem is referenced by:  bnj1018g  33632
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