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Theorem sbnfc2 4326
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 3402 . . . . 5 𝑦 ∈ V
2 csbtt 3807 . . . . 5 ((𝑦 ∈ V ∧ 𝑥𝐴) → 𝑦 / 𝑥𝐴 = 𝐴)
31, 2mpan 690 . . . 4 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝐴)
4 vex 3402 . . . . 5 𝑧 ∈ V
5 csbtt 3807 . . . . 5 ((𝑧 ∈ V ∧ 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝐴)
64, 5mpan 690 . . . 4 (𝑥𝐴𝑧 / 𝑥𝐴 = 𝐴)
73, 6eqtr4d 2776 . . 3 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
87alrimivv 1935 . 2 (𝑥𝐴 → ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
9 nfv 1921 . . 3 𝑤𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
10 eleq2 2821 . . . . . 6 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → (𝑤𝑦 / 𝑥𝐴𝑤𝑧 / 𝑥𝐴))
11 sbsbc 3684 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴[𝑦 / 𝑥]𝑤𝐴)
12 sbcel2 4305 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
1311, 12bitri 278 . . . . . 6 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
14 sbsbc 3684 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴[𝑧 / 𝑥]𝑤𝐴)
15 sbcel2 4305 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1614, 15bitri 278 . . . . . 6 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1710, 13, 163bitr4g 317 . . . . 5 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
18172alimi 1819 . . . 4 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
19 sbnf2 2359 . . . 4 (Ⅎ𝑥 𝑤𝐴 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
2018, 19sylibr 237 . . 3 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → Ⅎ𝑥 𝑤𝐴)
219, 20nfcd 2887 . 2 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑥𝐴)
228, 21impbii 212 1 (𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1540   = wceq 1542  wnf 1790  [wsb 2074  wcel 2114  wnfc 2879  Vcvv 3398  [wsbc 3680  csb 3790
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 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-nul 4212
This theorem is referenced by:  eusvnf  5259
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