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Theorem csbabg 2901
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 𝑉A / x{yφ} = {y[A / x]φ})
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 2827 . . . 4 ([z / y][A / x]φ[A / x][z / y]φ)
2 df-clab 2024 . . . . 5 (z {y[A / x]φ} ↔ [z / y][A / x]φ)
3 sbsbc 2762 . . . . 5 ([z / y][A / x]φ[z / y][A / x]φ)
42, 3bitri 173 . . . 4 (z {y[A / x]φ} ↔ [z / y][A / x]φ)
5 df-clab 2024 . . . . . 6 (z {yφ} ↔ [z / y]φ)
6 sbsbc 2762 . . . . . 6 ([z / y]φ[z / y]φ)
75, 6bitri 173 . . . . 5 (z {yφ} ↔ [z / y]φ)
87sbcbii 2812 . . . 4 ([A / x]z {yφ} ↔ [A / x][z / y]φ)
91, 4, 83bitr4i 201 . . 3 (z {y[A / x]φ} ↔ [A / x]z {yφ})
10 sbcel2g 2865 . . 3 (A 𝑉 → ([A / x]z {yφ} ↔ z A / x{yφ}))
119, 10syl5rbb 182 . 2 (A 𝑉 → (z A / x{yφ} ↔ z {y[A / x]φ}))
1211eqrdv 2035 1 (A 𝑉A / x{yφ} = {y[A / x]φ})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  [wsb 1642  {cab 2023  [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:  csbsng  3422  csbunig  3579  csbxpg  4364  csbdmg  4472  csbrng  4725
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