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Theorem bnj121 31457
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 3689 . 2 ([1𝑜 / 𝑛]𝜁[1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
3 bnj121.2 . 2 (𝜁′[1𝑜 / 𝑛]𝜁)
4 bnj105 31310 . . . . . . . 8 1𝑜 ∈ V
54bnj90 31308 . . . . . . 7 ([1𝑜 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1𝑜)
65bicomi 216 . . . . . 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 1195 . . . . 5 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
10 sbc3an 3691 . . . . 5 ([1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
119, 10bitr4i 270 . . . 4 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211imbi2i 328 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓)))
13 nfv 2010 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1413sbc19.21g 3698 . . . 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 270 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
172, 3, 163bitr4i 295 1 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  Vcvv 3385  [wsbc 3633   Fn wfn 6096  1𝑜c1o 7792   FrSe w-bnj15 31278
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 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035
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-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-pw 4351  df-sn 4369  df-suc 5947  df-fn 6104  df-1o 7799
This theorem is referenced by:  bnj150  31463  bnj153  31467
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