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Theorem bnj121 32413
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj121.1 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
bnj121.2 (𝜁′[1o / 𝑛]𝜁)
bnj121.3 (𝜑′[1o / 𝑛]𝜑)
bnj121.4 (𝜓′[1o / 𝑛]𝜓)
Assertion
Ref Expression
bnj121 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (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)   𝜁(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)   𝜓′(𝑥,𝑓,𝑛)   𝜁′(𝑥,𝑓,𝑛)

Proof of Theorem bnj121
StepHypRef Expression
1 bnj121.1 . . 3 (𝜁 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
21sbcbii 3736 . 2 ([1o / 𝑛]𝜁[1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
3 bnj121.2 . 2 (𝜁′[1o / 𝑛]𝜁)
4 bnj105 32265 . . . . . . . 8 1o ∈ V
54bnj90 32263 . . . . . . 7 ([1o / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1o)
65bicomi 227 . . . . . 6 (𝑓 Fn 1o[1o / 𝑛]𝑓 Fn 𝑛)
7 bnj121.3 . . . . . 6 (𝜑′[1o / 𝑛]𝜑)
8 bnj121.4 . . . . . 6 (𝜓′[1o / 𝑛]𝜓)
96, 7, 83anbi123i 1156 . . . . 5 ((𝑓 Fn 1o𝜑′𝜓′) ↔ ([1o / 𝑛]𝑓 Fn 𝑛[1o / 𝑛]𝜑[1o / 𝑛]𝜓))
10 sbc3an 3745 . . . . 5 ([1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1o / 𝑛]𝑓 Fn 𝑛[1o / 𝑛]𝜑[1o / 𝑛]𝜓))
119, 10bitr4i 281 . . . 4 ((𝑓 Fn 1o𝜑′𝜓′) ↔ [1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211imbi2i 339 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓)))
13 nfv 1920 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1413sbc19.21g 3752 . . . 4 (1o ∈ V → ([1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓))))
154, 14ax-mp 5 . . 3 ([1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1o / 𝑛](𝑓 Fn 𝑛𝜑𝜓)))
1612, 15bitr4i 281 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)) ↔ [1o / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
172, 3, 163bitr4i 306 1 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1o𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088  wcel 2113  Vcvv 3397  [wsbc 3679   Fn wfn 6328  1oc1o 8117   FrSe w-bnj15 32233
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229
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 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-pw 4487  df-sn 4514  df-suc 6172  df-fn 6336  df-1o 8124
This theorem is referenced by:  bnj150  32419  bnj153  32423
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