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Theorem sbnfc2 3060
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 2689 . . . . 5 𝑦 ∈ V
2 csbtt 3014 . . . . 5 ((𝑦 ∈ V ∧ 𝑥𝐴) → 𝑦 / 𝑥𝐴 = 𝐴)
31, 2mpan 420 . . . 4 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝐴)
4 vex 2689 . . . . 5 𝑧 ∈ V
5 csbtt 3014 . . . . 5 ((𝑧 ∈ V ∧ 𝑥𝐴) → 𝑧 / 𝑥𝐴 = 𝐴)
64, 5mpan 420 . . . 4 (𝑥𝐴𝑧 / 𝑥𝐴 = 𝐴)
73, 6eqtr4d 2175 . . 3 (𝑥𝐴𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
87alrimivv 1847 . 2 (𝑥𝐴 → ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
9 nfv 1508 . . 3 𝑤𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴
10 eleq2 2203 . . . . . 6 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → (𝑤𝑦 / 𝑥𝐴𝑤𝑧 / 𝑥𝐴))
11 sbsbc 2913 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴[𝑦 / 𝑥]𝑤𝐴)
12 sbcel2g 3023 . . . . . . . 8 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴))
131, 12ax-mp 5 . . . . . . 7 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
1411, 13bitri 183 . . . . . 6 ([𝑦 / 𝑥]𝑤𝐴𝑤𝑦 / 𝑥𝐴)
15 sbsbc 2913 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴[𝑧 / 𝑥]𝑤𝐴)
16 sbcel2g 3023 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴))
174, 16ax-mp 5 . . . . . . 7 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1815, 17bitri 183 . . . . . 6 ([𝑧 / 𝑥]𝑤𝐴𝑤𝑧 / 𝑥𝐴)
1910, 14, 183bitr4g 222 . . . . 5 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
20192alimi 1432 . . . 4 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
21 sbnf2 1956 . . . 4 (Ⅎ𝑥 𝑤𝐴 ↔ ∀𝑦𝑧([𝑦 / 𝑥]𝑤𝐴 ↔ [𝑧 / 𝑥]𝑤𝐴))
2220, 21sylibr 133 . . 3 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴 → Ⅎ𝑥 𝑤𝐴)
239, 22nfcd 2276 . 2 (∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑥𝐴)
248, 23impbii 125 1 (𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1329   = wceq 1331  wnf 1436  wcel 1480  [wsb 1735  wnfc 2268  Vcvv 2686  [wsbc 2909  csb 3003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004
This theorem is referenced by:  eusvnf  4377
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