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Theorem sbnfc2 3196
 Description: Two ways of expressing "x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbnfc2 (xAyz[y / x]A = [z / x]A)
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 2862 . . . . 5 y V
2 csbtt 3148 . . . . 5 ((y V xA) → [y / x]A = A)
31, 2mpan 651 . . . 4 (xA[y / x]A = A)
4 vex 2862 . . . . 5 z V
5 csbtt 3148 . . . . 5 ((z V xA) → [z / x]A = A)
64, 5mpan 651 . . . 4 (xA[z / x]A = A)
73, 6eqtr4d 2388 . . 3 (xA[y / x]A = [z / x]A)
87alrimivv 1632 . 2 (xAyz[y / x]A = [z / x]A)
9 nfv 1619 . . 3 wyz[y / x]A = [z / x]A
10 eleq2 2414 . . . . . 6 ([y / x]A = [z / x]A → (w [y / x]Aw [z / x]A))
11 sbsbc 3050 . . . . . . 7 ([y / x]w A ↔ [̣y / xw A)
12 sbcel2g 3157 . . . . . . . 8 (y V → ([̣y / xw Aw [y / x]A))
131, 12ax-mp 5 . . . . . . 7 ([̣y / xw Aw [y / x]A)
1411, 13bitri 240 . . . . . 6 ([y / x]w Aw [y / x]A)
15 sbsbc 3050 . . . . . . 7 ([z / x]w A ↔ [̣z / xw A)
16 sbcel2g 3157 . . . . . . . 8 (z V → ([̣z / xw Aw [z / x]A))
174, 16ax-mp 5 . . . . . . 7 ([̣z / xw Aw [z / x]A)
1815, 17bitri 240 . . . . . 6 ([z / x]w Aw [z / x]A)
1910, 14, 183bitr4g 279 . . . . 5 ([y / x]A = [z / x]A → ([y / x]w A ↔ [z / x]w A))
20192alimi 1560 . . . 4 (yz[y / x]A = [z / x]Ayz([y / x]w A ↔ [z / x]w A))
21 sbnf2 2108 . . . 4 (Ⅎx w Ayz([y / x]w A ↔ [z / x]w A))
2220, 21sylibr 203 . . 3 (yz[y / x]A = [z / x]A → Ⅎx w A)
239, 22nfcd 2484 . 2 (yz[y / x]A = [z / x]AxA)
248, 23impbii 180 1 (xAyz[y / x]A = [z / x]A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
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