Theorem csbvarg 4382

Description: The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbvarg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)

Proof of Theorem csbvarg

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3455	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3849	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥}
3		sbcel2gv 3806	. . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
4	3	eqabcdv 2863	. . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦)
5	2, 4	eqtrid 2777	. . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦)
6	5	elv 3439	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
7	6	csbeq2i 3856	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦
8		csbcow 3863	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥
9		df-csb 3849	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦}
10	7, 8, 9	3eqtr3i 2761	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦}
11		sbcel2gv 3806	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
12	11	eqabcdv 2863	. . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴)
13	10, 12	eqtrid 2777	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
14	1, 13	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2110  {cab 2708  Vcvv 3434  [wsbc 3739  ⦋csb 3848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3436  df-sbc 3740  df-csb 3849

This theorem is referenced by:  csbvargi  4383  sbccsb2  4385  2nreu  4392  csbfv  6864  ixpsnval  8819  csbwrdg  14443  swrdspsleq  14565  prmgaplem7  16961  telgsums  19898  ixpsnbasval  21135  scmatscm  22421  pm2mpf1lem  22702  pm2mpcoe1  22708  idpm2idmp  22709  pm2mpmhmlem2  22727  monmat2matmon  22732  pm2mp  22733  fvmptnn04if  22757  chfacfscmulfsupp  22767  cayhamlem4  22796  divcncf  25368  opsbc2ie  32445  esum2dlem  34095  relowlpssretop  37377  rdgeqoa  37383  renegclALT  38981  cdlemk40  40935  tfsconcatfv  43353  iscard4  43545  minregex  43546  cotrclrcl  43754  frege124d  43773  frege70  43945  frege72  43947  frege77  43952  frege91  43966  frege92  43967  frege116  43991  frege118  43993  frege120  43995  rusbcALT  44450  onfrALTlem5  44554  onfrALTlem4  44555  onfrALTlem5VD  44896  iccelpart  47443  ply1mulgsumlem4  48400