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Theorem bnj130 31273
Description: Technical lemma for bnj151 31276. 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 3633 . 2 ([1𝑜 / 𝑛]𝜃[1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
3 bnj130.4 . 2 (𝜃′[1𝑜 / 𝑛]𝜃)
4 bnj105 31121 . . . . . . . . . 10 1𝑜 ∈ V
54bnj90 31119 . . . . . . . . 9 ([1𝑜 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1𝑜)
65bicomi 214 . . . . . . . 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 1159 . . . . . . 7 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
10 sbc3an 3636 . . . . . . 7 ([1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
119, 10bitr4i 267 . . . . . 6 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211eubii 2630 . . . . 5 (∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
134bnj89 31118 . . . . 5 ([1𝑜 / 𝑛]∃!𝑓(𝑓 Fn 𝑛𝜑𝜓) ↔ ∃!𝑓[1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1412, 13bitr4i 267 . . . 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 1993 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1716sbc19.21g 3644 . . . 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 267 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))
202, 3, 193bitr4i 292 1 (𝜃′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2140  ∃!weu 2608  Vcvv 3341  [wsbc 3577   Fn wfn 6045  1𝑜c1o 7724   FrSe w-bnj15 31089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-pw 4305  df-sn 4323  df-suc 5891  df-fn 6053  df-1o 7731
This theorem is referenced by:  bnj151  31276
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