Theorem csbidm 4413

Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csbidm	⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csbidm

Step	Hyp	Ref	Expression
1		csbnest1g 4412	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵)
2		csbconstg 3898	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐴 = 𝐴)
3	2	csbeq1d 3883	. . 3 ⊢ (𝐴 ∈ V → ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	1, 3	eqtrd 2771	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
5		csbprc 4389	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ∅)
6		csbprc 4389	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
7	5, 6	eqtr4d 2774	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
8	4, 7	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ⦋csb 3879  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-nul 4314

This theorem is referenced by: (None)