Theorem csbvarg 4391

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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3847	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥}
3		sbcel2gv 3804	. . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
4	3	eqabcdv 2894	. . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦)
5	2, 4	eqtrid 2807	. . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦)
6	5	elv 3455	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
7	6	csbeq2i 3854	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦
8		csbcow 3861	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥
9		df-csb 3847	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦}
10	7, 8, 9	3eqtr3i 2791	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦}
11		sbcel2gv 3804	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
12	11	eqabcdv 2894	. . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴)
13	10, 12	eqtrid 2807	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
14	1, 13	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2738  Vcvv 3450  [wsbc 3738  ⦋csb 3846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sbc 3739  df-csb 3847

This theorem is used by:  csbvargi  4392  sbccsb2  4394  2nreu  4401  csbfv  6920  ixpsnval  8906  csbwrdg  14657  swrdspsleq  14783  prmgaplem7  17197  telgsums  20169  ixpsnbasval  21445  scmatscm  22790  pm2mpf1lem  23074  pm2mpcoe1  23080  idpm2idmp  23081  pm2mpmhmlem2  23099  monmat2matmon  23104  pm2mp  23105  fvmptnn04if  23129  chfacfscmulfsupp  23139  cayhamlem4  23168  divcncf  25730  opsbc2ie  33006  esum2dlem  34658  relowlpssretop  38207  rdgeqoa  38213  renegclALT  39940  cdlemk40  41894  tfsconcatfv  44286  iscard4  44477  minregex  44478  cotrclrcl  44686  frege124d  44705  frege70  44877  frege72  44879  frege77  44884  frege91  44898  frege92  44899  frege116  44923  frege118  44925  frege120  44927  rusbcALT  45366  onfrALTlem5  45469  onfrALTlem4  45470  onfrALTlem5VD  45811  iccelpart  48437  ply1mulgsumlem4  49423