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Theorem bnj985v 32504
Description: Version of bnj985 32505 with an additional disjoint variable condition, not requiring ax-13 2372. (Contributed by Gino Giotto, 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 32316 . . 3 𝐺 ∈ V
3 bnj985v.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj985v.11 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj984 32503 . . 3 (𝐺 ∈ V → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
62, 5ax-mp 5 . 2 (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒)
7 sbcex2 3743 . . 3 ([𝐺 / 𝑓]𝑝𝜒′ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
8 nfv 1921 . . . . . . 7 𝑝𝜒
98sb8ev 2356 . . . . . 6 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
10 sbsbc 3684 . . . . . . 7 ([𝑝 / 𝑛]𝜒[𝑝 / 𝑛]𝜒)
1110exbii 1854 . . . . . 6 (∃𝑝[𝑝 / 𝑛]𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
129, 11bitri 278 . . . . 5 (∃𝑛𝜒 ↔ ∃𝑝[𝑝 / 𝑛]𝜒)
13 bnj985v.6 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
1412, 13bnj133 32276 . . . 4 (∃𝑛𝜒 ↔ ∃𝑝𝜒′)
1514sbcbii 3738 . . 3 ([𝐺 / 𝑓]𝑛𝜒[𝐺 / 𝑓]𝑝𝜒′)
16 bnj985v.9 . . . 4 (𝜒″[𝐺 / 𝑓]𝜒′)
1716exbii 1854 . . 3 (∃𝑝𝜒″ ↔ ∃𝑝[𝐺 / 𝑓]𝜒′)
187, 15, 173bitr4i 306 . 2 ([𝐺 / 𝑓]𝑛𝜒 ↔ ∃𝑝𝜒″)
196, 18bitri 278 1 (𝐺𝐵 ↔ ∃𝑝𝜒″)
Colors of variables: wff setvar class
Syntax hints:  wb 209  w3a 1088   = wceq 1542  wex 1786  [wsb 2074  wcel 2114  {cab 2716  wrex 3054  Vcvv 3398  [wsbc 3680  cun 3841  {csn 4516  cop 4522   Fn wfn 6334  w-bnj17 32235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-uni 4797  df-bnj17 32236
This theorem is referenced by:  bnj1018  32515
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