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Theorem bnj130 31454
Description: Technical lemma for bnj151 31457. 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 3688 . 2 ([1𝑜 / 𝑛]𝜃[1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3 bnj130.4 . 2 (𝜃′[1𝑜 / 𝑛]𝜃)
4 bnj105 31303 . . . . . . . . . 10 1𝑜 ∈ V
54bnj90 31301 . . . . . . . . 9 ([1𝑜 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1𝑜)
65bicomi 216 . . . . . . . 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 1195 . . . . . . 7 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
10 sbc3an 3690 . . . . . . 7 ([1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
119, 10bitr4i 270 . . . . . 6 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211eubii 2625 . . . . 5 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
134bnj89 31300 . . . . 5 ([1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1412, 13bitr4i 270 . . . 4 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓))
1514imbi2i 328 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
16 nfv 2010 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1716sbc19.21g 3697 . . . 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 270 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
202, 3, 193bitr4i 295 1 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  ∃!weu 2608  Vcvv 3384  [wsbc 3632   Fn wfn 6095  1𝑜c1o 7791   FrSe w-bnj15 31271
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-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pow 5034
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-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-pw 4350  df-sn 4368  df-suc 5946  df-fn 6103  df-1o 7798
This theorem is referenced by:  bnj151  31457
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