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Theorem bnj121 31245
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 3630 . 2 ([1𝑜 / 𝑛]𝜁[1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
3 bnj121.2 . 2 (𝜁′[1𝑜 / 𝑛]𝜁)
4 bnj105 31097 . . . . . . . 8 1𝑜 ∈ V
54bnj90 31095 . . . . . . 7 ([1𝑜 / 𝑛]𝑓 Fn 𝑛𝑓 Fn 1𝑜)
65bicomi 214 . . . . . 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 1159 . . . . 5 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
10 sbc3an 3633 . . . . 5 ([1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓) ↔ ([1𝑜 / 𝑛]𝑓 Fn 𝑛[1𝑜 / 𝑛]𝜑[1𝑜 / 𝑛]𝜓))
119, 10bitr4i 267 . . . 4 ((𝑓 Fn 1𝑜𝜑′𝜓′) ↔ [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓))
1211imbi2i 325 . . 3 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → [1𝑜 / 𝑛](𝑓 Fn 𝑛𝜑𝜓)))
13 nfv 1990 . . . . 5 𝑛(𝑅 FrSe 𝐴𝑥𝐴)
1413sbc19.21g 3641 . . . 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 267 . 2 (((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)) ↔ [1𝑜 / 𝑛]((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 𝑛𝜑𝜓)))
172, 3, 163bitr4i 292 1 (𝜁′ ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → (𝑓 Fn 1𝑜𝜑′𝜓′)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2137  Vcvv 3338  [wsbc 3574   Fn wfn 6042  1𝑜c1o 7720   FrSe w-bnj15 31065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-pw 4302  df-sn 4320  df-suc 5888  df-fn 6050  df-1o 7727
This theorem is referenced by:  bnj150  31251  bnj153  31255
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