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Theorem csbopabg 4637
 Description: Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
csbopabg (A V[A / x]{y, z φ} = {y, z A / xφ})
Distinct variable groups:   y,z,A   x,y,z
Allowed substitution hints:   φ(x,y,z)   A(x)   V(x,y,z)

Proof of Theorem csbopabg
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3139 . . 3 (w = A[w / x]{y, z φ} = [A / x]{y, z φ})
2 dfsbcq2 3049 . . . 4 (w = A → ([w / x]φ ↔ [̣A / xφ))
32opabbidv 4625 . . 3 (w = A → {y, z [w / x]φ} = {y, z A / xφ})
41, 3eqeq12d 2367 . 2 (w = A → ([w / x]{y, z φ} = {y, z [w / x]φ} ↔ [A / x]{y, z φ} = {y, z A / xφ}))
5 vex 2862 . . 3 w V
6 nfs1v 2106 . . . 4 x[w / x]φ
76nfopab 4627 . . 3 x{y, z [w / x]φ}
8 sbequ12 1919 . . . 4 (x = w → (φ ↔ [w / x]φ))
98opabbidv 4625 . . 3 (x = w → {y, z φ} = {y, z [w / x]φ})
105, 7, 9csbief 3177 . 2 [w / x]{y, z φ} = {y, z [w / x]φ}
114, 10vtoclg 2914 1 (A V[A / x]{y, z φ} = {y, z A / xφ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046  [csb 3136  {copab 4622 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-3an 936  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  df-opab 4623 This theorem is referenced by: (None)
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