Theorem csbvarg 3164

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 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		vex 2863	. . . . . 6 ⊢ y ∈ V
3		df-csb 3138	. . . . . . 7 ⊢ [y / x]x = {z ∣ [̣y / x]̣z ∈ x}
4		sbcel2gv 3107	. . . . . . . 8 ⊢ (y ∈ V → ([̣y / x]̣z ∈ x ↔ z ∈ y))
5	4	eqabcdv 2470	. . . . . . 7 ⊢ (y ∈ V → {z ∣ [̣y / x]̣z ∈ x} = y)
6	3, 5	syl5eq 2397	. . . . . 6 ⊢ (y ∈ V → [y / x]x = y)
7	2, 6	ax-mp 5	. . . . 5 ⊢ [y / x]x = y
8	7	csbeq2i 3163	. . . 4 ⊢ [A / y][y / x]x = [A / y]y
9		csbco 3146	. . . 4 ⊢ [A / y][y / x]x = [A / x]x
10		df-csb 3138	. . . 4 ⊢ [A / y]y = {z ∣ [̣A / y]̣z ∈ y}
11	8, 9, 10	3eqtr3i 2381	. . 3 ⊢ [A / x]x = {z ∣ [̣A / y]̣z ∈ y}
12		sbcel2gv 3107	. . . 4 ⊢ (A ∈ V → ([̣A / y]̣z ∈ y ↔ z ∈ A))
13	12	eqabcdv 2470	. . 3 ⊢ (A ∈ V → {z ∣ [̣A / y]̣z ∈ y} = A)
14	11, 13	syl5eq 2397	. 2 ⊢ (A ∈ V → [A / x]x = A)
15	1, 14	syl 15	1 ⊢ (A ∈ V → [A / x]x = A)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  [̣wsbc 3047  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  sbccsb2g  3166  csbfvg  5339