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Theorem bnj985v 34921
Description: Version of bnj985 34922 with an additional disjoint variable condition, not requiring ax-13 2374. (Contributed by GG, 27-Mar-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj985v.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj985v.6 (𝜒′[𝑝 / 𝑛]𝜒)
bnj985v.9 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj985v.11 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj985v.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj985v (𝐺𝐵 ↔ ∃𝑝𝜒″)
Distinct variable groups:   𝐺,𝑝   𝜒,𝑝   𝑓,𝑝   𝑛,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 bnj985v
StepHypRef Expression
1 bnj985v.13 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
21bnj918 34734 . . 3 𝐺 ∈ V
3 bnj985v.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj985v.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj984 34920 . . 3 (𝐺 ∈ V → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
62, 5ax-mp 5 . 2 (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒)
7 sbcex2 3863 . . 3 ([𝐺 / 𝑓]𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8 nfv 1913 . . . . . . 7 𝑝𝜒
98sb8ef 2354 . . . . . 6 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10 sbsbc 3802 . . . . . . 7 ([𝑝 / 𝑛]𝜒[𝑝 / 𝑛]𝜒)
1110exbii 1846 . . . . . 6 (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
129, 11bitri 275 . . . . 5 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13 bnj985v.6 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
1412, 13bnj133 34695 . . . 4 (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
1514sbcbii 3859 . . 3 ([𝐺 / 𝑓]𝑛𝜒[𝐺 / 𝑓]𝑝𝜒′)
16 bnj985v.9 . . . 4 (𝜒″[𝐺 / 𝑓]𝜒′)
1716exbii 1846 . . 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 206  w3a 1087   = wceq 1537  wex 1777  [wsb 2064  wcel 2103  {cab 2711  wrex 3072  Vcvv 3482  [wsbc 3798  cun 3968  {csn 4648  cop 4654   Fn wfn 6567  w-bnj17 34654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3073  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-sn 4649  df-pr 4651  df-uni 4932  df-bnj17 34655
This theorem is referenced by:  bnj1018  34932
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