Theorem csbvarg 3077

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 2741	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		vex 2733	. . . . . 6 ⊢ 𝑦 ∈ V
3		df-csb 3050	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥}
4		sbcel2gv 3018	. . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
5	4	abbi1dv 2290	. . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦)
6	3, 5	eqtrid 2215	. . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦)
7	2, 6	ax-mp 5	. . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦
8	7	csbeq2i 3076	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦
9		csbco 3059	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥
10		df-csb 3050	. . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦}
11	8, 9, 10	3eqtr3i 2199	. . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦}
12		sbcel2gv 3018	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
13	12	abbi1dv 2290	. . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴)
14	11, 13	eqtrid 2215	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
15	1, 14	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  {cab 2156  Vcvv 2730  [wsbc 2955  ⦋csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  sbccsb2g  3079  csbfvg  5534  f1od2  6214  bj-sels  13949