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Theorem bnj985 32234
Description: Technical lemma for bnj69 32290. 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 32233 for a version with more disjoint variable conditions, not requiring ax-13 2390. (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 32045 . . 3 𝐺 ∈ V
3 bnj985.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj985.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj984 32232 . . 3 (𝐺 ∈ V → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
62, 5ax-mp 5 . 2 (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒)
7 sbcex2 3813 . . 3 ([𝐺 / 𝑓]𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8 nfv 1915 . . . . . . 7 𝑝𝜒
98sb8e 2560 . . . . . 6 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10 sbsbc 3756 . . . . . . 7 ([𝑝 / 𝑛]𝜒[𝑝 / 𝑛]𝜒)
1110exbii 1848 . . . . . 6 (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
129, 11bitri 277 . . . . 5 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13 bnj985.6 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
1412, 13bnj133 32005 . . . 4 (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
1514sbcbii 3808 . . 3 ([𝐺 / 𝑓]𝑛𝜒[𝐺 / 𝑓]𝑝𝜒′)
16 bnj985.9 . . . 4 (𝜒″[𝐺 / 𝑓]𝜒′)
1716exbii 1848 . . 3 (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
187, 15, 173bitr4i 305 . 2 ([𝐺 / 𝑓]𝑛𝜒 ↔ ∃𝑝𝜒″)
196, 18bitri 277 1 (𝐺𝐵 ↔ ∃𝑝𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wb 208  w3a 1083   = wceq 1537  wex 1780  [wsb 2069  wcel 2114  {cab 2798  wrex 3126  Vcvv 3473  [wsbc 3752  cun 3911  {csn 4543  cop 4549   Fn wfn 6326  w-bnj17 31964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rex 3131  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-sn 4544  df-pr 4546  df-uni 4815  df-bnj17 31965
This theorem is referenced by:  bnj1018g  32243
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