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Theorem csbabg 3197
 Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
csbabg (A V[A / x]{y φ} = {y A / xφ})
Distinct variable groups:   y,A   x,y
Allowed substitution hints:   φ(x,y)   A(x)   V(x,y)

Proof of Theorem csbabg
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbccom 3117 . . . 4 ([̣z / y]̣[̣A / xφ ↔ [̣A / x]̣[̣z / yφ)
2 df-clab 2340 . . . . 5 (z {y A / xφ} ↔ [z / y][̣A / xφ)
3 sbsbc 3050 . . . . 5 ([z / y][̣A / xφ ↔ [̣z / y]̣[̣A / xφ)
42, 3bitri 240 . . . 4 (z {y A / xφ} ↔ [̣z / y]̣[̣A / xφ)
5 df-clab 2340 . . . . . 6 (z {y φ} ↔ [z / y]φ)
6 sbsbc 3050 . . . . . 6 ([z / y]φ ↔ [̣z / yφ)
75, 6bitri 240 . . . . 5 (z {y φ} ↔ [̣z / yφ)
87sbcbii 3101 . . . 4 ([̣A / xz {y φ} ↔ [̣A / x]̣[̣z / yφ)
91, 4, 83bitr4i 268 . . 3 (z {y A / xφ} ↔ [̣A / xz {y φ})
10 sbcel2g 3157 . . 3 (A V → ([̣A / xz {y φ} ↔ z [A / x]{y φ}))
119, 10syl5rbb 249 . 2 (A V → (z [A / x]{y φ} ↔ z {y A / xφ}))
1211eqrdv 2351 1 (A V[A / x]{y φ} = {y A / xφ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  csbsng  3785  csbunig  3899  csbxpg  4813  csbrng  4966
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