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Theorem bnj985 31540
Description: Technical lemma for bnj69 31595. 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
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 31353 . . 3 𝐺 ∈ V
3 bnj985.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj985.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj984 31539 . . 3 (𝐺 ∈ V → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
62, 5ax-mp 5 . 2 (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒)
7 sbcex2 3684 . . 3 ([𝐺 / 𝑓]𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8 nfv 2010 . . . . . . 7 𝑝𝜒
98sb8e 2543 . . . . . 6 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10 sbsbc 3637 . . . . . . 7 ([𝑝 / 𝑛]𝜒[𝑝 / 𝑛]𝜒)
1110exbii 1944 . . . . . 6 (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
129, 11bitri 267 . . . . 5 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13 bnj985.6 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
1412, 13bnj133 31313 . . . 4 (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
1514sbcbii 3689 . . 3 ([𝐺 / 𝑓]𝑛𝜒[𝐺 / 𝑓]𝑝𝜒′)
16 bnj985.9 . . . 4 (𝜒″[𝐺 / 𝑓]𝜒′)
1716exbii 1944 . . 3 (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
187, 15, 173bitr4i 295 . 2 ([𝐺 / 𝑓]𝑛𝜒 ↔ ∃𝑝𝜒″)
196, 18bitri 267 1 (𝐺𝐵 ↔ ∃𝑝𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wb 198  w3a 1108   = wceq 1653  wex 1875  [wsb 2064  wcel 2157  {cab 2785  wrex 3090  Vcvv 3385  [wsbc 3633  cun 3767  {csn 4368  cop 4374   Fn wfn 6096  w-bnj17 31272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-nul 4116  df-sn 4369  df-pr 4371  df-uni 4629  df-bnj17 31273
This theorem is referenced by:  bnj1018  31549
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