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Theorem sbnfc2 4151
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 3344 . . . . 5 𝑦 ∈ V
2 csbtt 3686 . . . . 5 ((𝑦 ∈ V ∧ 𝑥𝐴) → 𝑦 / 𝑥𝐴 = 𝐴)
31, 2mpan 708 . . . 4 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝐴)
4 vex 3344 . . . . 5 𝑧 ∈ V
5 csbtt 3686 . . . . 5 ((𝑧 ∈ V ∧ 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝐴)
64, 5mpan 708 . . . 4 (𝑥𝐴𝑧 / 𝑥𝐴 = 𝐴)
73, 6eqtr4d 2798 . . 3 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
87alrimivv 2006 . 2 (𝑥𝐴 → ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
9 nfv 1993 . . 3 𝑤𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
10 eleq2 2829 . . . . . 6 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → (𝑤𝑦 / 𝑥𝐴𝑤𝑧 / 𝑥𝐴))
11 sbsbc 3581 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴[𝑦 / 𝑥]𝑤𝐴)
12 sbcel2 4133 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
1311, 12bitri 264 . . . . . 6 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
14 sbsbc 3581 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴[𝑧 / 𝑥]𝑤𝐴)
15 sbcel2 4133 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1614, 15bitri 264 . . . . . 6 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1710, 13, 163bitr4g 303 . . . . 5 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
18172alimi 1889 . . . 4 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
19 sbnf2 2577 . . . 4 (Ⅎ𝑥 𝑤𝐴 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
2018, 19sylibr 224 . . 3 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → Ⅎ𝑥 𝑤𝐴)
219, 20nfcd 2898 . 2 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑥𝐴)
228, 21impbii 199 1 (𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1630   = wceq 1632  wnf 1857  [wsb 2047  wcel 2140  wnfc 2890  Vcvv 3341  [wsbc 3577  csb 3675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-nul 4060
This theorem is referenced by:  eusvnf  5011
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