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Theorem csbabg 2884
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 2810 . . . 4 ([z / y][A / x]φ[A / x][z / y]φ)
2 df-clab 2009 . . . . 5 (z {y[A / x]φ} ↔ [z / y][A / x]φ)
3 sbsbc 2745 . . . . 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 2009 . . . . . 6 (z {yφ} ↔ [z / y]φ)
6 sbsbc 2745 . . . . . 6 ([z / y]φ[z / y]φ)
75, 6bitri 173 . . . . 5 (z {yφ} ↔ [z / y]φ)
87sbcbii 2795 . . . 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 2848 . . 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 2020 1 (A 𝑉A / x{yφ} = {y[A / x]φ})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  [wsb 1627  {cab 2008  [wsbc 2741  csb 2829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-csb 2830
This theorem is referenced by:  csbsng  3405  csbunig  3562  csbxpg  4348  csbdmg  4456  csbrng  4709
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