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Theorem bnj130 30198
 Description: Technical lemma for bnj151 30201. 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
bnj130.1 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
bnj130.2 (𝜑′[1𝑜 / 𝑛]𝜑)
bnj130.3 (𝜓′[1𝑜 / 𝑛]𝜓)
bnj130.4 (𝜃′[1𝑜 / 𝑛]𝜃)
Assertion
Ref Expression
bnj130 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝜓(𝑥,𝑓,𝑛)   𝜃(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)   𝜓′(𝑥,𝑓,𝑛)   𝜃′(𝑥,𝑓,𝑛)

Proof of Theorem bnj130
StepHypRef Expression
1 bnj130.1 . . 3 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
21sbcbii 3458 . 2 ([1𝑜 / 𝑛]𝜃[1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3 bnj130.4 . 2 (𝜃′[1𝑜 / 𝑛]𝜃)
4 bnj105 30044 . . . . . . . . . 10 1𝑜 ∈ V
54bnj90 30042 . . . . . . . . 9 ([1𝑜 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1𝑜)
65bicomi 213 . . . . . . . 8 (𝑓 Fn 1𝑜[1𝑜 / 𝑛]𝑓 Fn 𝑛)
7 bnj130.2 . . . . . . . 8 (𝜑′[1𝑜 / 𝑛]𝜑)
8 bnj130.3 . . . . . . . 8 (𝜓′[1𝑜 / 𝑛]𝜓)
96, 7, 83anbi123i 1244 . . . . . . 7 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
10 sbc3an 3461 . . . . . . 7 ([1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
119, 10bitr4i 266 . . . . . 6 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211eubii 2480 . . . . 5 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
134bnj89 30041 . . . . 5 ([1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1412, 13bitr4i 266 . . . 4 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
1514imbi2i 325 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
16 nfv 1830 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1716sbc19.21g 3469 . . . 4 (1𝑜 ∈ V → ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))))
184, 17ax-mp 5 . . 3 ([1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
1915, 18bitr4i 266 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
202, 3, 193bitr4i 291 1 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402   Fn wfn 5799  1𝑜c1o 7440   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-suc 5646  df-fn 5807  df-1o 7447 This theorem is referenced by:  bnj151  30201
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