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Theorem sbnfc2 4201
Description: Two ways of expressing "𝑥 is (effectively) not free in 𝐴". (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbnfc2 (𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbnfc2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 vex 3386 . . . . 5 𝑦 ∈ V
2 csbtt 3737 . . . . 5 ((𝑦 ∈ V ∧ 𝑥𝐴) → 𝑦 / 𝑥𝐴 = 𝐴)
31, 2mpan 682 . . . 4 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝐴)
4 vex 3386 . . . . 5 𝑧 ∈ V
5 csbtt 3737 . . . . 5 ((𝑧 ∈ V ∧ 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝐴)
64, 5mpan 682 . . . 4 (𝑥𝐴𝑧 / 𝑥𝐴 = 𝐴)
73, 6eqtr4d 2834 . . 3 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
87alrimivv 2024 . 2 (𝑥𝐴 → ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
9 nfv 2010 . . 3 𝑤𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
10 eleq2 2865 . . . . . 6 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → (𝑤𝑦 / 𝑥𝐴𝑤𝑧 / 𝑥𝐴))
11 sbsbc 3635 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴[𝑦 / 𝑥]𝑤𝐴)
12 sbcel2 4182 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
1311, 12bitri 267 . . . . . 6 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
14 sbsbc 3635 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴[𝑧 / 𝑥]𝑤𝐴)
15 sbcel2 4182 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1614, 15bitri 267 . . . . . 6 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1710, 13, 163bitr4g 306 . . . . 5 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
18172alimi 1908 . . . 4 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
19 sbnf2 2558 . . . 4 (Ⅎ𝑥 𝑤𝐴 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
2018, 19sylibr 226 . . 3 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → Ⅎ𝑥 𝑤𝐴)
219, 20nfcd 2934 . 2 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑥𝐴)
228, 21impbii 201 1 (𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1651   = wceq 1653  wnf 1879  [wsb 2064  wcel 2157  wnfc 2926  Vcvv 3383  [wsbc 3631  csb 3726
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 2375  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-nul 4114
This theorem is referenced by:  eusvnf  5060
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