Theorem csbidmg 3191

Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)

Assertion

Ref	Expression
csbidmg	⊢ (A ∈ V → [A / x][A / x]B = [A / x]B)

Distinct variable group:   x,A

Allowed substitution hints:   B(x)   V(x)

Proof of Theorem csbidmg

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		csbnest1g 3189	. . 3 ⊢ (A ∈ V → [A / x][A / x]B = [[A / x]A / x]B)
3		csbconstg 3151	. . . 4 ⊢ (A ∈ V → [A / x]A = A)
4	3	csbeq1d 3143	. . 3 ⊢ (A ∈ V → [[A / x]A / x]B = [A / x]B)
5	2, 4	eqtrd 2385	. 2 ⊢ (A ∈ V → [A / x][A / x]B = [A / x]B)
6	1, 5	syl 15	1 ⊢ (A ∈ V → [A / x][A / x]B = [A / x]B)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [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-3an 936  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: (None)