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Theorem csbopabg 3826
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 𝑉A / x{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [A / x]φ})
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 2849 . . 3 (w = Aw / x{⟨y, z⟩ ∣ φ} = A / x{⟨y, z⟩ ∣ φ})
2 dfsbcq2 2761 . . . 4 (w = A → ([w / x]φ[A / x]φ))
32opabbidv 3814 . . 3 (w = A → {⟨y, z⟩ ∣ [w / x]φ} = {⟨y, z⟩ ∣ [A / x]φ})
41, 3eqeq12d 2051 . 2 (w = A → (w / x{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [w / x]φ} ↔ A / x{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [A / x]φ}))
5 vex 2554 . . 3 w V
6 nfs1v 1812 . . . 4 x[w / x]φ
76nfopab 3816 . . 3 x{⟨y, z⟩ ∣ [w / x]φ}
8 sbequ12 1651 . . . 4 (x = w → (φ ↔ [w / x]φ))
98opabbidv 3814 . . 3 (x = w → {⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [w / x]φ})
105, 7, 9csbief 2885 . 2 w / x{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [w / x]φ}
114, 10vtoclg 2607 1 (A 𝑉A / x{⟨y, z⟩ ∣ φ} = {⟨y, z⟩ ∣ [A / x]φ})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  [wsb 1642  [wsbc 2758  csb 2846  {copab 3808
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-3an 886  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  df-opab 3810
This theorem is referenced by:  csbcnvg  4462
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