Theorem csbidm 4399

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 4398	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵)
2		csbconstg 3873	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐴 = 𝐴)
3	2	csbeq1d 3858	. . 3 ⊢ (𝐴 ∈ V → ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	1, 3	eqtrd 2798	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
5		csbprc 4375	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ∅)
6		csbprc 4375	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
7	5, 6	eqtr4d 2801	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
8	4, 7	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⦋csb 3854  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-nul 4288

This theorem is referenced by: (None)