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Theorem sbnfc2 2900
Description: Two ways of expressing "x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbnfc2 (xAyzy / 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 2554 . . . . 5 y V
2 csbtt 2856 . . . . 5 ((y V xA) → y / xA = A)
31, 2mpan 400 . . . 4 (xAy / xA = A)
4 vex 2554 . . . . 5 z V
5 csbtt 2856 . . . . 5 ((z V xA) → z / xA = A)
64, 5mpan 400 . . . 4 (xAz / xA = A)
73, 6eqtr4d 2072 . . 3 (xAy / xA = z / xA)
87alrimivv 1752 . 2 (xAyzy / xA = z / xA)
9 nfv 1418 . . 3 wyzy / xA = z / xA
10 eleq2 2098 . . . . . 6 (y / xA = z / xA → (w y / xAw z / xA))
11 sbsbc 2762 . . . . . . 7 ([y / x]w A[y / x]w A)
12 sbcel2g 2865 . . . . . . . 8 (y V → ([y / x]w Aw y / xA))
131, 12ax-mp 7 . . . . . . 7 ([y / x]w Aw y / xA)
1411, 13bitri 173 . . . . . 6 ([y / x]w Aw y / xA)
15 sbsbc 2762 . . . . . . 7 ([z / x]w A[z / x]w A)
16 sbcel2g 2865 . . . . . . . 8 (z V → ([z / x]w Aw z / xA))
174, 16ax-mp 7 . . . . . . 7 ([z / x]w Aw z / xA)
1815, 17bitri 173 . . . . . 6 ([z / x]w Aw z / xA)
1910, 14, 183bitr4g 212 . . . . 5 (y / xA = z / xA → ([y / x]w A ↔ [z / x]w A))
20192alimi 1342 . . . 4 (yzy / xA = z / xAyz([y / x]w A ↔ [z / x]w A))
21 sbnf2 1854 . . . 4 (Ⅎx w Ayz([y / x]w A ↔ [z / x]w A))
2220, 21sylibr 137 . . 3 (yzy / xA = z / xA → Ⅎx w A)
239, 22nfcd 2170 . 2 (yzy / xA = z / xAxA)
248, 23impbii 117 1 (xAyzy / xA = z / xA)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  [wsb 1642  wnfc 2162  Vcvv 2551  [wsbc 2758  csb 2846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by:  eusvnf  4151
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