Theorem csbidmg 3105

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 2741	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		csbnest1g 3104	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵)
3		csbconstg 3063	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐴 = 𝐴)
4	3	csbeq1d 3056	. . 3 ⊢ (𝐴 ∈ V → ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
5	2, 4	eqtrd 2203	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
6	1, 5	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  Vcvv 2730  ⦋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-3an 975  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: (None)